Quiz Practice - Random Variables

Instructions:

Complete the following exercises and enter your answers into Blackboard. Hardcopy answers will not be accepted.

Let $X$ and $Y$ be independent random variables with:

• ${\mu }_X=9.5$, ${\sigma }_X=0.4$ (mean and standard deviation of $X$)
• ${\mu }_Y=6.8$, ${\sigma }_Y=0.1$ (mean and standard deviation of $Y$)

Question 1: Mean of $3X$

Formula for the mean of aX:

${\mu }_{aX}=a{\mu }_X$

For this problem, $a=3$ and ${\mu }_X=9.5$: ${\mu }_{3X}=3\times 9.5=28.5$

Answer: $28.5$

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Question 2: Standard Deviation of $3X$

Formula for the standard deviation of aX:

${\sigma }_{aX}=|a|{\sigma }_X$

For this problem, $a=3$ and ${\sigma }_X=0.4$: ${\sigma }_{3X}=3\times 0.4=1.2$

Answer: $1.2$

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Question 3: Mean of $Y-X$

Formula for the mean of Y - X:

${\mu }_{Y-X}={\mu }_Y-{\mu }_X$

Substitute the given values ${\mu }_Y=6.8$ and ${\mu }_X=9.5$: ${\mu }_{Y-X}=6.8-9.5=-2.7$

Answer: $-2.7$

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Question 4: Standard Deviation of $Y-X$

Formula for the standard deviation of Y - X:

Since $X$ and $Y$ are independent, the variance of the difference is: ${\sigma }_{Y-X}^2={\sigma }_Y^2+{\sigma }_X^2$ The standard deviation is the square root of the variance: ${\sigma }_{Y-X}=\sqrt{{\sigma }_Y^2+{\sigma }_X^2}$

Substitute the given values ${\sigma }_Y=0.1$ and ${\sigma }_X=0.4$: ${\sigma }_{Y-X}=\sqrt{(0.1{)}^2+(0.4{)}^2}=\sqrt{0.01+0.16}=\sqrt{0.17}\approx 0.4123$

Answer: $0.4123$

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Question 5: Mean of $X+4Y$

Formula for the mean of X + 4Y:

${\mu }_{X+4Y}={\mu }_X+4{\mu }_Y$

Substitute ${\mu }_X=9.5$ and ${\mu }_Y=6.8$: ${\mu }_{X+4Y}=9.5+4\times 6.8=9.5+27.2=36.7$

Answer: $36.7$

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Question 6: Standard Deviation of $X+4Y$

Formula for the standard deviation of X + 4Y:

For independent random variables, the variance is: ${\sigma }_{X+4Y}^2={\sigma }_X^2+(4{\sigma }_Y{)}^2$ The standard deviation is: ${\sigma }_{X+4Y}=\sqrt{{\sigma }_X^2+(4{\sigma }_Y{)}^2}$

Substitute ${\sigma }_X=0.4$ and ${\sigma }_Y=0.1$: ${\sigma }_{X+4Y}=\sqrt{(0.4{)}^2+(4\times 0.1{)}^2}=\sqrt{0.16+0.16}=\sqrt{0.32}\approx 0.5657$

Answer: $0.5657$