Question

The length in inches of a cricket chosen at random from a field is a random variable $X$with mean $1.2$inches and standard deviation of $0.25$ inches. Find the mean and standard deviation of the length $Y$ of a randomly chosen cricket from the field in centimeters. There are $2.54$ centimeters in an inch.

Short answer

The mean and standard deviation are:

${\mu }_Y=3.048$

${\sigma }_Y=0.635$

Step-by-step solution

Given Information

Given

${\mu }_X=1.2$

${\sigma }_X=0.25$

$2.5\mathrm{cm}=1$inch

Since $1$inch is also $2.54\mathrm{cm}$:

$Y=2.54X$

Explanation

Properties mean and standard deviation:

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

${\sigma }_{aX+b}=a{\sigma }_X$

Then we obtain for $Y=2.54X$:

localid="1649911083440" $$\begin{gathered} {\mu }_Y={\mu }_{2.54X} \\ =2.54{\mu }_X \\ =2.54(1.2) \\ =3.048 \end{gathered}$$

localid="1649911096910" $$\begin{gathered} {\sigma }_Y={\sigma }_{2.54X} \\ =2.54{\sigma }_X \\ =2.54(0.25) \\ =0.635 \end{gathered}$$