2022-11-27

When Sam goes to a restaurant, he always tips the sener $2 plus 10% of the cost of the meal. If Sam's distribution of meal costs has a mean of$9 and a standard deviation of \$3, what are the mean and standard deviation of the distribution

skilningugdK

Expert

Let X be the random variable which represents the meal costs
Then ,

The random variable representing the Tips is 2 + 10% of X i.e; 2+0.1X.
Then,
$E\left(0.1X+2\right)=0.1E\left(X\right)+2=0.1\left(9\right)+2=2.9$
and
$\text{Var}\left(0.1X+2\right)=\left(0.1{\right)}^{2}\text{Var}\left(X\right)=\left(0.01\right)\left(9\right)=0.09\phantom{\rule{0ex}{0ex}}⇒sd\left(0.1X+2\right)=\sqrt{\text{Var(0.1X+2)}}=\sqrt{0.09}=0.3$

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