The amount of time it takes to serve each customer in a bank is a random variable,X,with a mean of ? minutes and a standard deviation of ? minutes.When you arrive at the bank there are three customers in front of you.If the times for the three customers are independent of one another,which of the following shows the correct method for calculating the variance of your wait time?

A) Var( $X _ { 1 }$ + $x _ { 2 }$ + $X _ { 3 }$ ) = $3 ^ { 2 }$ Var(X) = 9 $σ2\sigma ^ { 2 }$
B) Var(3X) = 3Var(X) = 3 $σ2\sigma ^ { 2 }$
C) Var( $X _ { 1 }$ + $X _ { 2 }$ + $X _ { 3 }$ ) = $3\sqrt { 3 }$ Var(X) = $3\sqrt { 3 }$ $σ2\sigma ^ { 2 }$
D) Var(3X) = $3 ^ { 2 }$ Var(X) = 9 $σ2\sigma ^ { 2 }$
E) Var( $X _ { 1 }$ + $X_ { 2 }$ + $X _ { 3 }$ ) = Var( $X _ { 1 }$ ) + Var( $X _ { 2 }$ ) + Var( $X _ { 3 }$ ) = 3 $σ2\sigma ^ { 2 }$

Question

The amount of time it takes to serve each customer in a bank is a random variable,X,with a mean of ? minutes and a standard deviation of ? minutes.When you arrive at the bank there are three customers in front of you.If the times for the three customers are independent of one another,which of the following shows the correct method for calculating the variance of your wait time?

A) Var(

X _ { 1 }

+

x _ { 2 }

+

X _ { 3 }

) =

3 ^ { 2 }

Var(X) = 9

σ2\sigma ^ { 2 }

B) Var(3X) = 3Var(X) = 3

σ2\sigma ^ { 2 }

C) Var(

X _ { 1 }

+

X _ { 2 }

+

X _ { 3 }

) =

3\sqrt { 3 }

Var(X) =

3\sqrt { 3 }

σ2\sigma ^ { 2 }

D) Var(3X) =

3 ^ { 2 }

Var(X) = 9

σ2\sigma ^ { 2 }

E) Var(

X _ { 1 }

+

X_ { 2 }

+

X _ { 3 }

) = Var(

X _ { 1 }

) + Var(

X _ { 2 }

) + Var(

X _ { 3 }

) = 3

σ2\sigma ^ { 2 }

Manuel hernandez · Accepted Answer

Final Answer : E, Explanation :   The variance of the sum of independent random variables is the sum of their variances. Since each customer's service time is independent and identically distributed with variance   σ 2 \sigma^2 σ 2  , the total variance is   3 σ 2 3\sigma^2 3 σ 2  .

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