The Amount of Money That Maria Earns in a Week $\$ 900$

The amount of money that Maria earns in a week is a random variable,X,with a mean of $\$ 900$ and a standard deviation of $\sigma _ { \mathrm { m } }$ .The amount of money that Daniel earns in a week is a random variable,Y,with a mean of $\$ 800$ and a standard deviation of $\sigma _ { \mathrm { d } }$ .The difference, $X - Y,$ between Maria's weekly income and Daniel's weekly income is a random variable with a mean of $\$ 900 - \$ 800 = \$ 100$ If the incomes are independent of one another,which of the following shows the correct method for calculating the standard deviation of the random variable $X - Y ?$

A) SD(X - Y) = Var(X - Y) = Var(X) + Var(Y) = $\sigma _ { \mathrm { m } } ^ { 2 }$ + $\sigma _ { \mathrm { d } } ^ { 2 }$
B) SD(X - Y) = $\sqrt { \operatorname { Var } ( X - Y ) }$ = $\sqrt { \operatorname { Var } ( X ) + \operatorname { Var } ( Y ) }$
= $\sqrt { \sigma _ { \mathrm { m } } ^ { 2 } + \sigma _ { \mathrm { d } } ^ { 2 } }$
C) SD(X - Y) = SD(X) + SD(Y) = $\sigma _ { \mathrm { m } }$ + $\sigma _ { \mathrm { d } }$
D) SD(X - Y) = SD(X) - SD(Y) = $\sigma _ { \mathrm { m } }$ - $\sigma _ { \mathrm { d } }$
E) SD(X - Y) = $\sqrt { \operatorname { Var } ( X - Y ) }$ = $\sqrt { \operatorname { Var } ( X ) - \operatorname { Var } ( Y ) }$
= $\sqrt { \sigma _ { \mathrm { m } } ^ { 2 } - \sigma _ { \mathrm { d } } ^ { 2 } }$

Correct Answer:

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