Q.4.37 - A First Course in Probability

Question

Find Var(X) and Var(Y) for X and Y as given in Problem 4.21 

Step-by-Step Solution

Answer

In the given information the answers are$Var\left(X\right)=80.92 ,Var\left(Y\right)=84.5$

Step 1:Given Information

$Var\left(x\right)=E\left(X^2\right)-E{\left(X\right)}^2$

Step 2:Calculation

By using the result d\from problem 22 and same notation, we can compute the variance using the identity $\mathrm{Var}\left(X\right)=E\left(X^2\right)-E(X{)}^2$.

Now we have that

$E\left(X^2\right)={40}^2·\frac{40}{148}+{33}^2·\frac{33}{148}+{25}^2·\frac{25}{148}+{50}^2·\frac{50}{148}=1625.42$

Then we have that,

$Var\left(X\right)=E\left(X^2\right)-E(X{)}^2=1625.42-39.3^2=80.92$

Similarly for $Y$,

$E\left(Y^2\right)={40}^2·\frac{1}{4}+{33}^2·\frac{1}{4}+{25}^2·\frac{1}{4}+{50}^2·\frac{1}{4}=1453.5$

Now, we have that

$Var\left(Y\right)=E\left(Y^2\right)-E(Y{)}^2=1453.5-{37}^2=84.5$

Step 3:Final Answer

The value of $Var\left(X\right)=80.92,Var\left(Y\right)=84.5$