Question

check your understanding
let ( x = ) the number billionaires in a randomly selected state. based on current records, the probability
distribution of ( x ) is as follows:

the random variable ( x ) has mean ( mu_x = 12.68 ) and standard deviation ( sigma_x = 29.02 ). suppose a law is passed
requiring each billionaire to pay a ( $ 7,000 ) tax to their state. let ( y = ) the tax money received by a randomly
selected state.

1. for a state that has 9 billionaires ( (x = 9) ), how much tax money (y) would they receive?
2. consider the graph of the probability distribution of ( x ) and a separate graph of the probability

distribution of ( y ). how would their shapes compare?

1. find the mean and standard deviation of ( y ).
2. each state agrees to invest ( $ 20,000 ) of the tax money they receive to improve roads. therefore,

the net amount (n) of money a randomly selected state has left to spend is ( n = y - $ 20,000 ).
describe the shape, mean, and standard deviation of the probability distribution of n.

Explanation:

1. Step - by - Step Format for Question 1:

• Explanation:
• Step1: Determine the relationship between \(X\) and \(Y\)
• We know that \(Y = 7000X\) (since each of the \(X\) billionaires pays a \(\$7000\) tax).
• Step2: Substitute \(X = 9\) into the equation
• When \(X = 9\), we substitute into \(Y=7000X\). So \(Y = 7000\times9\).
• \(Y=63000\).
• Answer:
• The tax money received when \(X = 9\) is \(\$63000\).

1. Answer - Explanation Format for Question 2:

• Brief Explanations:
• Since \(Y = 7000X\), a linear transformation of the form \(Y=aX\) (where \(a = 7000>0\)) does not change the shape of the probability distribution. A linear transformation \(Y=aX + b\) (in this case \(b = 0\)) only changes the scale (and location if \(b

eq0\)) of the distribution.

• Answer:
• The shapes of the probability distributions of \(X\) and \(Y\) are the same.

1. Step - by - Step Format for Question 3:

• Explanation:
• Step1: Use the formula for the mean of a linear transformation
• The formula for the mean of a linear transformation \(Y=aX + b\) (here \(b = 0\)) is \(\mu_Y=a\mu_X\). Given \(a = 7000\) and \(\mu_X=12.68\), we have \(\mu_Y=7000\times12.68\).
• \(\mu_Y = 7000\times12.68=88760\).
• Step2: Use the formula for the standard deviation of a linear transformation
• The formula for the standard deviation of a linear transformation \(Y=aX + b\) (where \(b = 0\)) is \(\sigma_Y=\vert a\vert\sigma_X\). Since \(a = 7000>0\) and \(\sigma_X = 29.02\), we have \(\sigma_Y=7000\times29.02\).
• \(\sigma_Y=7000\times29.02 = 203140\).
• Answer:
• The mean of \(Y\) is \(\mu_Y=\$88760\) and the standard deviation of \(Y\) is \(\sigma_Y=\$203140\).

1. Answer - Explanation Format for Question 4:

• Brief Explanations:
• Shape:
• Since \(N=Y - 20000\) (a linear transformation \(N=aY + b\) with \(a = 1\) and \(b=-20000\)), the shape of the probability distribution of \(N\) is the same as the shape of the probability distribution of \(Y\) (and thus the same as the shape of the probability distribution of \(X\)).
• Mean:
• Using the formula for the mean of a linear transformation \(\mu_N=\mu_Y-20000\). Since \(\mu_Y = 88760\), then \(\mu_N=88760-20000=68760\).
• Standard deviation:
• Using the formula for the standard deviation of a linear transformation \(\sigma_N=\sigma_Y\) (because for \(N = Y-20000\), \(a = 1\) in \(N=aY + b\)). So \(\sigma_N = 203140\).
• Answer:
• Shape: The same as the shape of the probability distribution of \(X\) (and \(Y\)). Mean: \(\mu_N=\$68760\). Standard deviation: \(\sigma_N=\$203140\).

Answer:

1. Step - by - Step Format for Question 1:

• Explanation:
• Step1: Determine the relationship between \(X\) and \(Y\)
• We know that \(Y = 7000X\) (since each of the \(X\) billionaires pays a \(\$7000\) tax).
• Step2: Substitute \(X = 9\) into the equation
• When \(X = 9\), we substitute into \(Y=7000X\). So \(Y = 7000\times9\).
• \(Y=63000\).
• Answer:
• The tax money received when \(X = 9\) is \(\$63000\).

1. Answer - Explanation Format for Question 2:

• Brief Explanations:
• Since \(Y = 7000X\), a linear transformation of the form \(Y=aX\) (where \(a = 7000>0\)) does not change the shape of the probability distribution. A linear transformation \(Y=aX + b\) (in this case \(b = 0\)) only changes the scale (and location if \(b

eq0\)) of the distribution.

• Answer:
• The shapes of the probability distributions of \(X\) and \(Y\) are the same.

1. Step - by - Step Format for Question 3:

• Explanation:
• Step1: Use the formula for the mean of a linear transformation
• The formula for the mean of a linear transformation \(Y=aX + b\) (here \(b = 0\)) is \(\mu_Y=a\mu_X\). Given \(a = 7000\) and \(\mu_X=12.68\), we have \(\mu_Y=7000\times12.68\).
• \(\mu_Y = 7000\times12.68=88760\).
• Step2: Use the formula for the standard deviation of a linear transformation
• The formula for the standard deviation of a linear transformation \(Y=aX + b\) (where \(b = 0\)) is \(\sigma_Y=\vert a\vert\sigma_X\). Since \(a = 7000>0\) and \(\sigma_X = 29.02\), we have \(\sigma_Y=7000\times29.02\).
• \(\sigma_Y=7000\times29.02 = 203140\).
• Answer:
• The mean of \(Y\) is \(\mu_Y=\$88760\) and the standard deviation of \(Y\) is \(\sigma_Y=\$203140\).

1. Answer - Explanation Format for Question 4:

• Brief Explanations:
• Shape:
• Since \(N=Y - 20000\) (a linear transformation \(N=aY + b\) with \(a = 1\) and \(b=-20000\)), the shape of the probability distribution of \(N\) is the same as the shape of the probability distribution of \(Y\) (and thus the same as the shape of the probability distribution of \(X\)).
• Mean:
• Using the formula for the mean of a linear transformation \(\mu_N=\mu_Y-20000\). Since \(\mu_Y = 88760\), then \(\mu_N=88760-20000=68760\).
• Standard deviation:
• Using the formula for the standard deviation of a linear transformation \(\sigma_N=\sigma_Y\) (because for \(N = Y-20000\), \(a = 1\) in \(N=aY + b\)). So \(\sigma_N = 203140\).
• Answer:
• Shape: The same as the shape of the probability distribution of \(X\) (and \(Y\)). Mean: \(\mu_N=\$68760\). Standard deviation: \(\sigma_N=\$203140\).