Using Area and perimeter formulae of rectangle,
a) the rate of change of Area is 14 cm²/sec
b) the rate of change of perimeter is zero.
c) length of diagonal is 13 cm
We have given that
length (l) is decreasing at the rate of 2 cm/min
and the width (w) is increasing at the rate of
2cm/min,
⇒ dl/dt = −2 cm/min and dw/dt = 2cm/sse
Thus the perimeter (P) of a rectangle is,
P=2(l+w) ---(1)
differentiating above equation with respect to time (t) we get,
dp/dt = 2(dl/dt + dw/dt )
=> dp/dt = 2( -2 + 2 ) = 0
=> P = constant
Hence, there is no change in perimeter of rectangle.
(b)Now, it is given that length (l) is decreasing at the rate of 2 cm/min
and the width (w) is increasing at the rate of 2 cm/min,
Thus the area (A) of a rectangle is, A= l× w
=> dA/dt = l ×dw/dt + w×dl/dt (used differentaion
we have l = 12 cm and w = 5 cm
=> dA/dt = 12 (2) + 5(-2)
=> dA/dt = 24 - 10 = 14
Hence, the area is increasing at the rate of
14 cm²/min.
c) length of diagonal = √(12)² + (5)² = √144+25
=> length of diagonal = √169 = 13 cm
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Which is not a power function?.
A function containing a variable base raised to a fixed power is considered to be a power function.
when a function has a constant base raised to a variable power. This can be called an exponential function, not a power function A power function may be a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. It is in the form of f(x)=kx^p, where k and p are the real numbers whereas k is the coefficient.
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F(x)= 12x-5 and G(x) = 2x²+3
Find (G-F) (-2) =
The domain of the function is 24.The collection of all potential inputs for a function is known as its domain.
How do you locate the domain?Let y = f(x) be a function with x as the independent variable and y as the dependent variable. A value for x is said to be in the domain of a function f if it can be used to successfully produce a single value, y, using another value for x.The collection of all potential inputs for a function is known as its domain. For instance, the domain of f(x)=x2 is all real numbers, and the domain of g(x)=1/x is all real numbers other than x=0.Determine the input values. Exclude any real numbers that produce a negative value in the radicand since there is an even root. Set the radicand to be bigger than or equal to Calculate x from zero. The area of the function where the solution(s) lie.Explanation:
F(x)= 12x-5 and G(x) = 2x²+3
Find (G-F) (-2) , put x = 0 , 2+3-12-5 = -12.
-12 * -2 = 24.
The domain of the function is 24.
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8. in the pcr process, if we assume that each cycle takes 5 minutes, how many fold amplification would be accomplished in an hour?
The answer to the given question is 4096-fold. If it assumed that each cycle in the PCR process takes 5 minutes, then total 4096-fold amplification could be accomplished in an hour.
What are the steps in PCR Amplification?PCR is based on 3 simple steps needed for any DNA synthesis reaction. Those steps are: denaturation of the template to the single strands; annealing of the primers to every original strand for a new strand synthesis; and development of the new DNA strands from the primers.
In this case, each cycle takes 5 minutes and doubles the DNA.
Thus, in one hour, there would be 60 / 5 = 12 cycles.
So, the DNA would be amplified 212 = 4096-fold.
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for each of the following, determine the direction in which the objective function increases: a z 4x1 x2 b z x1 2x2 c z x1 3x2
The direction of increase of the given objective functions is
(a) in the direction of x₁
(b) in the direction of x₂
(c) in the direction of x₂
In the question ,
three objective functions are given ,
we have to find the direction in which the given function increase ,
So ,
Part (a) ,
z = 4x₁ - x₂
According to the signs of the variables given ,
the function z will increase in the direction of x₁ , because x₁ has positive sign and it will decrease in the direction of x₂ as it has a negative sign .
Part(b) ,
z = -x₁ + 2x₂
According to the signs of the variables given ,
the function z will increase in the direction of x₂ , because x₂ has positive sign and it will decrease in the direction of x₁ as it has a negative sign .
Part(c) ,
z = -x₁ + 3x₂
According to the signs of the variables given ,
the function z will increase in the direction of x₂ , because x₂ has positive sign and it will decrease in the direction of x₁ as it has a negative sign .
Therefore , The direction of increase of the given objective functions is
(a) in the direction of x₁
(b) in the direction of x₂
(c) in the direction of x₂ .
The given question is incomplete , the complete question is
For each of the following, determine the direction in which the objective function increases:
(a) z = 4x₁ - x₂
(b) z = -x₁ + 2x₂
(c) z = -x₁ + 3x₂
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Jan traveled 260 miles using 8 gallons of gas. What was the car’s miles per quart on this trip?
Answer:
130 mi/ per quart
Step-by-step explanation:
260 mi/8 gallons =32.5 mi/gallon
1 gallon = 4 quarts, so 32.5 gallons is 130 quarts
tom is a big green bay packers and aaron rodgers fan. he claims that aaron rodger's average number of touchdowns per game is 2.45. dave (a hardcore bears fan) believes that tom might be inflating rodgers's statistics. from his debut until november 18, 2019, aaron rodgers has played 175 games! dave does not want to look at all of them. he randomly selects 52 games that rodgers played and finds the sample average number of touchdowns to be 1.942 with a standard deviation of 1.42. do a hypothesis test and state your conclusion. use the significance level
After doing a hypothesis test the conclusion is reject [tex]H_0[/tex] .
Given:
tom is a big green bay packers and aaron rodgers fan. he claims that aaron rodger's average number of touchdowns per game is 2.45. dave (a hardcore bears fan) believes that tom might be inflating rodgers's statistics. from his debut until november 18, 2019, aaron rodgers has played 175 games! dave does not want to look at all of them. he randomly selects 52 games that rodgers played and finds the sample average number of touchdowns to be 1.942 with a standard deviation of 1.42.
here n = 52
σ = 1.42
x = 1.942
μ = 2.45
z = x - μ / σ/√n
= 1.942 - 2.45 / 1.42/√52
= -2.58
p value = 0.0049
[tex]H_0:mean = 2.45[/tex]
[tex]H_a:mean < 2.45[/tex]
Rejection rule:Reject [tex]H_0[/tex] if p value < 0.05
Conclusion Reject [tex]H_0[/tex]
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In a random series, successive observations are probabilistically independent of one another. If this property is violated, the observations are said to be:
If this property is violated, the observations are said to be: autocorrelated
What is Random Series?
The results of an experiment are functions of an integer argument, and the results are described by a random series: U1, U2.. (or, simply, sequences of random values - 1st value, 2nd value, etc).
The term "time series" and, occasionally, "random processes with discrete time" are frequently used to describe random series. The Markov chain exemplifies a crucial category of random series.
In this question, it's asked when the property of independence is violated.
If the consecutive observations are correlated or linearly related, then they are said to be autocorrelated. This denotes that there will be an association of the current time period with the successive and preceding time periods. Violation of the autocorrelation assumption yields inaccurate results.
Hence, The observations are said to be: autocorrelated in the random series.
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Put these numbers in order from greatest to least.
-0.85
12.45
0.25
-12
Answer:
12.45, 0.25, -0.85, -12
Step-by-step explanation:
The larger the negative the lesser the value, opposite for positive numbers
What is the solution to the system of equations y x 10 2x y 4?.
The solution to the system of equation is [tex](\frac{14}{3},-\frac{16}{3})[/tex]
What is a system of equations ?A system of equations is a set of two or more equations with the same variables. A solution to a system of equations is a set of values for the variable that satisfy all the equations simultaneously.
Differentiate between substitution and elimination method.
The major difference between the substitution and elimination method is that the substitution method is the process of replacing the variable with a value, whereas the elimination method is the process of removing the variable from the system of linear equations.
The system of equations are :
y = x - 10 and
2x + y = 4
On solving two equations by using substitution method ,we get
By substituting the value of y in another equation ,we obtain the equation in terms of one variable :
Now, according to the question
⇒ 2x + (x - 10) = 4
⇒ (2x + x) = (4 + 10)
⇒ 3x = 14
⇒ x = 14/3
Further for y ,we substitute obtained value of x in the equation
y = x - 10
Now,
[tex]y = \frac{14}{3}-10[/tex]
[tex]=\frac{(14-30)}{3}[/tex]
[tex]= (\frac{-16}{3})[/tex]
[tex]= - (\frac{16}{3})[/tex]
Hence ,the solution is (x,y) : [tex](\frac{14}{3},-\frac{16}{3})[/tex]
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If the measures, in degrees, of the three angles of a triangle are x, x + 10, and 2x - 6, the triangle must be isosceles equilateral right scalene
The triangle with angles 44 degrees, 54 degrees, and 82 degrees will be a scalene triangle.
We are given the following:
The measures, in degrees, of the three angles of a triangle are x, x + 10, and 2x - 6.
We know from the triangle angle sum property that the sum of its internal angles is 180 degrees.
Let us find the given angles;
x + x + 10 + 2x - 6 = 180 degrees
4x + 4 = 180 degrees
4x = 180 - 4 degrees
x = 176 / 4 degrees
x = 44 degrees
So,
Angles will be;
x = 44 degrees
x + 10 = 44 + 10 = 54 degrees
2x - 6 = 2 * 44 - 6 = 88 - 6 = 82 degrees
All the angles are different.
Thus, the triangle with angles 44 degrees, 54 degrees, and 82 degrees will be a scalene triangle.
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You are a member of the fund-raiing committee for your chool. The committee i ponoring a fair. Each member of the committee ha agreed to ell 50 ticket to the fair. Student ticket are $3 each and adult ticket are $5 each. Let t repreent the number of tudent ticket that you ell. Write and implify an expreion in term of t for the total amount of money you raie for elling all 50 ticket. Then ue the expreion to determine how much money you will raie if you ell all 50 ticket and 28 of thoe are tudent ticket
The expression that represents the total amount that can be raised from selling 50 tickets is -2t + 250 and the total amount that can be raised is $194.
What is total revenue?
Total revenue is the amount of money earned by a company from the sale of its goods and services. In other words, businesses use this metric to determine how well their core revenue-generating operations are performing. Total revenue is directly related to marginal revenue.
Number of tickets that will be sold = 50
Number of student tickets that will be sold = t
Number of adult tickets that will be sold = 50 - t
The total amount of money that can be raised from selling 50 tickets is the sum of the total amount raised from selling student tickets and adult tickets.
Total revenue = total amount raised from selling students tickets + total amount raised from selling adults tickets
Total revenue = [3 x t] + [(50 -t) x $5]
Total revenue = $3t + $250 - 5t
Total revenue = -2t + 250
Total revenue = -2(28) + 250
Total revenue = -56 + 250 = $194
Hence, the expression that represents the total amount that can be raised from selling 50 tickets is -2t + 250, and the total amount that can be raised is $194.
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a drawer contains 20 socks. 10 are black. 6 are blue. 4 are red. two socks are taken out at random. what is the probability that one is blue and one is red?
Chance is also referred to as probability and the probability that one is blue and one is red is 4/19
probability = Number of event/Total samples space
If there are 10 black socks,20 white socks,2 red socks, and 6 blue socks in a drawer, the total number of socks will be the sample space
Sample space = 10+20+2+6
Sample space = 38
Total number of blue and red socks is the event
Number of event one blue and one red = 8 blue socks
Chance that the socks is blue = n(E)/n(S)
Chance that the socks is one red and one blue = 8/38
Chance that the socks is red and blue = 4/19
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Government data show that 26% of the civilian labor force has at least 4 years of college and that 15% of the labor force works as laborers or operators of machines or vehicles. Can you conclude that because (0.26)(0.15) = .039 about 4% of the labor force are college-educated laborers or operators?
(a) Yes, by the multiplication rule
(b) Yes, by conditional probabilities
(c) Yes,bythelawoflargenumbers
(d) No, because the events are not independent
(e) No, because the events are not mutually exclusive
d
Option (d)- No, because the events are not independent. Hence we can not conclude that because (0.26)(0.15) = 0.039 about 4% of the labor force are college-educated labors or operators.
We have given that government data show that 26% of the civilian labor force has at least 4 years of college and that 15% of the labor force works as laborers or operators of machine or vehicle.
So here,
event A = 26% of civilian labor force hast at least 4 years of college
event B = 15% of the labor force works as laborers or operators of machines or vehicles
Therefore it is observe that both the events are not independent.
Hence we cannot conclude that because (0.26)(0.15) = 0.039 about 4% of the labor force are college-educated laborers or operators.
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a sector of a circle has a central angle of $100^\circ$. if the area of the sector is $250 \pi$, what is the radius of the circle?
The required radius of given circle is 30 units.
What is circle and area of its sector?
A circle is a shut two-layered figure wherein the arrangement of the multitude of focuses in the plane is equidistant from a given point called "focus". Each line that goes through the circle shapes the line of reflection evenness. Likewise, it has rotational evenness around the middle for each point.
Area of sector of circle:Picture result for What is circle and region of its area?
The equation for the region of the area of a circle is(r2) /360where r is the span of the circle and is the point of the area.
According to question:Angle of sector = 100 degree, area= 250π
Let radius of circle be r
Then,
area of sector = (r^2) /360
250π = (r2) 100/360
r^2 = 250×360/100
r = √25×36 = 30 units
Thus, radius of the circle is 30units.
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write 60:52 in the form n:1.
Converted form of ratio 60:52 in form n:1 is 10/7 : 1.
How to write a ratio in n:1 form?
In order that a ratio is written in the form n:1 we must make the right hand side equal to one.
n can be expressed as a decimal or a fraction.
According to the given question:
Convert ratio 60:52 in n:1 form
So, divide both sides of ratio with 52 to get 1 on right side of ratio.
60/52 : 1
10/7 : 1
Hence 10/7 : 1 is the converted form.
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How do you graph a linear equation that passes through two points?.
The equation of a line that passes through points (a,b) and (c,d) is (c-a)y = (d-b)x+b(c-a)-a(d-b).
What is equation of line?A line's equation is an algebraic way of expressing the collection of points that make up a line in a coordinate system.
The many points that collectively make up a line on the coordinate axis are represented as a group of variables (x, y) to create an algebraic equation, also known as an equation of a line.
Let the two points are (a,b) and (c, d)
To find the equation of line,
Use formula y- y₁ = (y₂-y₁/x₂-x₁) (x-x₁)
The equation of line,
y-b = (d-b/c-a)(x-a)
(c-a)(y-b) = (d-b)(x-a)
(c-a)y = (d-b)x+b(c-a)-a(d-b)
The required equation of the line is (c-a)y = (d-b)x+b(c-a)-a(d-b).
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helpppppppppppppppppppppp plsssss..
Answer:
The function crosses the Y axis (the up down axis) at -1
Step-by-step explanation:
A boat traveled 27 miles in 2 hrs. At this rate, How many miles will the boat travele in 1/2 hrs
Answer:
In explanation down below.
Step-by-step explanation:
So if the boat was able to travel 27 miles in 2 hours that means that it traveled 13.5 miles an hour. I think that you want to know how many miles the boat will travel in 1/2 an hour. Which is 6.75.
1 hour: 13.5 miles
1/2 hour the boat has traveled 6.75 miles.
I hope this helps!!!
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in a complete undirected graph with five vertices how many cells will contain a value of 1 in an adjacency matrix?
There are 5 1's in the given identity matrix of order 5.
Given, an identity matrix of order 5
we have to find the number of 1 in the given matrix
So, as the matrix is an identity matrix and the order of the matrix is 5
in an identity matrix, the entry is 1 when
i = j i.e. on the diagonal elements.
As, the order of the matrix is 5 then, the number of 1's in the matrix are 5.
Hence, there are 5 1's in the given identity matrix of order 5.
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Lin wrote a proof to show that diagonal EG is a
line of symmetry for rhombus EFGH. Fill in the
blanks to complete her proof.
Because EFGH is a rhombus, the distance from E to F is the same distance from E to H. Since E is the same distance from F as it is from H, it must lie on the perpendicular bisector of FH. By the same reasoning, G must lie on the perpendicular bisector of FH. Therefore, line EG is the perpendicular bisector of segment FH. So, reflecting rhombus EFGH across line FH will take E to G and G to E (because E and G are on the line of reflection) and F to H and H to F (since FH is perpendicular to the line of reflection, and R and H are the same distance from the line of reflection, on opposite sides). Since the image of rhombus EFGH reflected across EG is rhombus EHGF (the same rhombus), line EG must be a line of symmetry from rhombus EFGH.
The function g(x) is a transformation of the parent function f(x). Decide how
f(x) was transformed to make g(x).
f(x)
g(x)
x
-2
-1
2
لیا
у
619
9
-13
9
27
81
-2
-1
2
3
4
9
3
110
27
81
OA. Horizontal or vertical stretch
B. Horizontal or vertical shift
C. Horizontal or vertical reflection
D. Reflection across the line y=x
f(x) was transformed to make g(x) by B) Horizontal or vertical stretch.
How does transformation of a function happens?The transformation of a function may involve any change.
Usually, these can be shift horizontally (by transforming inputs) or vertically (by transforming output), stretching (multiplying outputs or inputs) etc.
Given that the function g(x) is a transformation of the parent function f(x).
To Decide how f(x) was transformed to make g(x).
The reflection of the point (x,y) across the line y = 1/2 x is the point (y, x).
By the given tables we can see that;
g(x) = 1/2 f (x)
Therefore, f(x) was transformed to make g(x) by B) Horizontal or vertical stretch.
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1) Four cards labeled A, B, C, and D are randomly placed in four boxes labeled A, B, C, and D. Each box receives exactly one card.
In how many ways can the cards be placed in the boxes?
_______ ways
2)four cards labeled A, B, C, and D are randomly placed in four boxes labeled A, B, C, and D. Each box receives exactly one card.
Count the number of elements in the event that no box contains a card with the same letter as the box.
________ elements
If no box has a card with the same letter as the box, then count the number of elements as 24 ways.
Given that,
Four randomly selected cards with the letters A, B, C, and D are placed in the corresponding boxes. There is exactly one card in each box.
We have to find If no box has a card with the same letter as the box, then count the number of elements.
We know that,
What is the permutation?A permutation is a mathematical calculation that establishes the number of possible arrangements for a given set, taking into account the arrangement's order.
From the given information:
The permutation approach can be used to represent the four cards that are randomly placed in the four boxes and can be used to express the following:
=⁴P₄
Since no box contains a card that begins with the same letter as the box, it follows that
Card A can be inserted into a box four different ways,
Card B three different ways,
Card C two ways, and
Card D one way.
The four cards' permutation can then be calculated as follows:
⁴P₄=4!/0!=4×3×2×1/1=24.
Therefore, If no box has a card with the same letter as the box, then count the number of elements as 24 ways.
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How many triangles exist with the given side lengths?.
Exactly one unique triangle exists with the given side lengths, option B is correct.
Here, we are given that the three sides of triangle are- 12, 15 and 18.
According to the triangle inequality theorem- “The sum of the length of any two sides of a triangle should be greater than the third side”. Let us see if this holds in this case-
(12 + 15) > 18
(15 + 18) > 12
(12 + 18) > 15
Thus, we see that the triangle inequality theorem is satisfied. So there is only one triangle possible with the sides of given lengths.
Hence, exactly one unique triangle exists with the given side lengths, option B is correct.
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Your question was incomplete. Check for full content below-
How many triangles exist with the given side lengths?
12 in, 15 in, 18 in
A) No triangle exists with the given side lengths.
B) Exactly one unique triangle exists with the given side lengths.
C) More than one triangle exists with the given side lengths.
Is mean absolute deviation the same as mean?.
The Mean absolute deviation and the mean are not same.
Given,
The mean;-
The average of a group of variables is referred to as the mean in mathematics and statistics. There are several methods for calculating the mean, including simple arithmetic means (adding the numbers together and dividing the result by the number of observations), geometric means, and harmonic means.
Mean absolute deviation;-
The average distance between each data point and the mean is known as the mean absolute deviation of a dataset. It offers us a sense of how variable a dataset contains.
Therefore,
The Mean absolute deviation and the mean are not same.
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If the first and second terms of a geometric sequence are 3 and 12, what is
the expression for the value of the 24th term of the sequence?
Answer:
[tex]3(4)^{23}[/tex]
Step-by-step explanation:
Let the first term be [tex]a[/tex] and the common ratio be [tex]r[/tex].
We know [tex]a=3[/tex] and [tex]ar=12[/tex], meaning [tex]r=4[/tex].
This means the 24th term is [tex]3(4)^{23}[/tex].
Aven wants to buy a car in 4 years and needs a down payment of $2500. If she deposits $2000
now, with interest compounded continuously, what interest rate will she need to get her down
payment in time?
She needs to get her interest rate at will 5.592 %/year.
What is Compound interest?
Compound interest is the addition of interest to the principal sum of a loan or deposit or interest on interest plus interest.
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
Where,
A = final amount
P = initial principal balance
r = interest rate
n = number of times interest is applied per time period
t = number of time periods elapsed
Solving for rate r as a decimal
r = n[(A/P)1/nt - 1]
r = 12 × [(2,500.00/2,000.00)1/(12)(4) - 1]
r = 0.0559158
Then convert r to R as a percentage
R = r * 100
R = 0.0559158 * 100
R = 5.592%/year
Hence, she needs to get her interest rate at will 5.592 %/year.
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What is an outlier in 6th grade math?.
If the data collection contains the given following numbers: 1, 19, 25, 32, 36, 38, 31, 42, 57, 45, and 87 , Here 1 and 87 represents the outlier.
What is characteristics of statistics ?The important characteristics of the Statistics that are given as follows:
Statistics have always been numerically expressed.
It had an aggregate of all the facts and dataData are to be collected in proper wayIt should be always comparable to one anotherData are collected for a well planned purposeOutliers are the data's most wildly divergent values.For instance, if the data collection contains the following numbers: 1, 19, 25, 32, 36, 38, 31, 42, 57, 45, and 87
The display of all data and outliers
The numbers 1 and 87, which are at the extreme and are hence outliers in the number line above, may be seen to be at these extremes.
The group outliers are a part of it, but they are not always close to the other group members.
Outliers are a concern as they cause an imbalance in the data set, which is why they are typically eliminated. Furthermore, an error can sometimes cause an outlier to appear in a data collection.
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Tulip bulbs are sold in packs of 25 or 40
lily buys a pack of 25 and kiki buys in a pack of 40
what is the smallest number of packs they can each buy so they have the same number of bulbs?
show working please, if solve you will be thanks and marked as the brainliest.
which of these statements is not true? if the average variance inflation factor is greater than 1 then the regression model might be biased. multicollinearity in the data is shown by a vif (variance inflation factor) greater than 10. tolerance values above 0.2 may indicate multicollinearity in the data. the tolerance is 1 divided by the vif (variance inflation factor).
Point C - Tolerance values above 0.2 may indicate multi collinearity in the data is correct.
Given that
It must be below 0.2, along with VIF above 10.
The remaining options are not correct.
A. If the average variance inflation factor is greater than 1 then the regression model might be biased.
B. Multi collinearity in the data is shown by a VIF (variance inflation factor) greater than 10.
D. The tolerance is 1 divided by the VIF (variance inflation factor).
Hence the answer is Point C - Tolerance values above 0.2 may indicate multi collinearity in the data is correct.
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in a casino, you play the following game: you choose a number between 1 and 6 and then throw 3 fair six-sided dice. after tossing, for each die showing your chosen number, the casino pays you $1. In order to play this game, you have to pay the casino a stake of S1 Let X denote your net win after one round of gambling (i.e., the payment received from the casino minus your stake) (a) Determine the state space Sx of X (b) Compute the probability mass function px of X (c) Compute the expected value E[X] of X (d) In general, a game is called a fair game if the net win of the gambler is 0 "on average" Is the above game a fair game? If not, what should be the stake of the gambler in order to make this a fair game? Give a reason for your answer.
By using the concept of probability, it can be calculated that
a) State space S(x) = {-1, 0, 1, 2}
b) The required probability mass function is
P(X = -1) = [tex]\frac{125}{216}[/tex]
P(X = 0) [tex]= \frac{25}{72}[/tex]
P(X = 1) [tex]= \frac{5}{72}[/tex]
P(X = 2) [tex]= \frac{1}{216}[/tex]
c) Expected value of X = -0.5
d) the stake of the gambler in order to make this a fair game is $0.5
What is probability?
Probability gives us the information about how likely an event is going to occur
Probability is calculated by Number of favourable outcomes divided by the total number of outcomes.
Probality of any event is greater than or equal to zero and less than or equal to 1.
Probability of sure event is 1 and probability of unsure event is 0.
a) Let a number n be chosen between 1 and 6
Three fair six sided dice are thrown
The following cases can occur
1) Number n occur on none of the three dice
2) Number n occur on one dice
3) Number n occur on two dice
4) Number n occur on three dice
Let X denote your net win after one round of gambling (i.e., the payment received from the casino minus your stake)
1) X = 0 -1 = -1
2) X = 1 - 1 = 0
3) X = 2 - 1 = 1
4) X = 3 - 1 = 2
State space S(x) = {-1, 0, 1, 2}
b) P(X = -1) = [tex]{3 \choose 0}(\frac{1}{6})^0(\frac{5}{6})^3 = (\frac{5}{6})^3 = \frac{125}{216}[/tex]
P(X = 0) = [tex]{3 \choose 1})(\frac{1}{6})(\frac{5}{6})^2 = \frac{25}{72}[/tex]
P(X = 1) = [tex]{3 \choose 2}(\frac{1}{6})^2\frac{5}{6} = \frac{5}{72}[/tex]
P(X = 2) = [tex]{3 \choose 3}(\frac{1}{6})^3(\frac{5}{6})^0 = \frac{1}{216}[/tex]
The required probability mass function is
P(X = -1) = [tex]\frac{125}{216}[/tex]
P(X = 0) [tex]= \frac{25}{72}[/tex]
P(X = 1) [tex]= \frac{5}{72}[/tex]
P(X = 2) [tex]= \frac{1}{216}[/tex]
c) Expected value of X = [tex]-1 \times \frac{125}{216} + 0 \times \frac{25}{72} + 1 \times \frac{5}{72} + 2\times \frac{1}{216}[/tex]
= [tex]\frac{-125 + 0 + 15 +2}{216}\\[/tex]
= [tex]-\frac{108}{125}[/tex]
= -0.5
d) The gambler looses $0.5 on a average
So this is not a fair game
Let the stake of the gambler in order to make this a fair game be t
By the problem,
-0.5 + t = 0
t = 0 + 0.5
t = $0.5
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