Find Var(X) and Var(Y) for X and Y as given in Problem 21.

Find Var(X) and Var(Y) for X and Y as given in Problem 21.

H
240 points

Find Var(X) and Var(Y) for X and Y as given in Problem 21

Find Var
hansel

8.7k points

Var(X)

since,

$$Var(X)=E(X^2)-(E(X))^2$$

we need to find $E(X^2)$. We found (E(X))^2 already in Problem 21.

$$E(X^2)=\sum_{i=1}^4 x_i^2 p(x_i)=40^2\bigg(\frac{40}{148}\bigg)+33^2\bigg(\frac{33}{148}\bigg)+25^2\bigg(\frac{25}{148}\bigg)+50^2\bigg(\frac{50}{148}\bigg)=1625.42$$

therefore,

$$Var(X)=E(X^2)-(E(X))^2=1625.42-(39.28)^2=82.50$$

Var(Y)

same reasoning as before,

$$E(Y^2)=\sum_{i=1}^4 y_i^2 p(y_i)=40^2\bigg(\frac{1}{4}\bigg)+33^2\bigg(\frac{1}{4}\bigg)+25^2\bigg(\frac{1}{4}\bigg)+50^2\bigg(\frac{1}{4}\bigg)=1453.50$$

and hence

$$Var(Y)=E(Y^2)-(E(Y))^2=1453.50-(37)^2=84.50$$

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