Q. A bag contains 5 white balls and 3 black balls. 3 balls are drawn at randomly without replacement. If X is random variable, which takes value 1 if at least 2 white balls are drawn, and value 0, otherwise, find E(X).
Ans: $\small X_1=1$ if the number of white balls are at least 2. $\small P(X_1 \geq 2)=\frac{^5C_2.^3C_1}{^8C_3}+\frac{^5C_3.^3C_0}{^8C_3}=\frac{40}{56}$
When
$\small X_1 : 1$
$\small P(X_1):\frac {40}{56}$
$\small X_2=0$ if the number of white balls are must be less than 2. $\small P(X_2 < 2)=\frac{^5C_1.^3C_2}{^8C_3}+\frac{^5C_0.^3C_3}{^8C_3}=\frac{16}{56}$
When
$\small X_2 : 0$
$\small P(X_2):\frac {16}{56}$
$\small E(X)=\sum_{i=1}^{2} X_iP(X_i)=1.\frac{40}{56}+0.\frac{16}{56}=\frac{5}{7}$
Q. If X and Y are both negative random variables, mutually independent. E(XY)=6 and |E(X)|=2, Find E(Y).
Ans: As we know X,Y are both negative and mutually independent so $\small E(XY)=E(X)E(Y)$
$\small E(X)E(Y)=6$ and $\small |E(X)|=2$ and we know X,Y are both negative random variable so,
$\small E(X)=-2$ and $\small E(Y)=\frac {E(XY)}{E(X)}=-\frac{6}{2}=-3$
Q.Show that $\small [E(X^2)]^{\frac{1}{2}} \geq E(X)$.
Ans: We know that $\small V(X) \geq 0$
or,$\small E(X^2)\geq [E(X)]^2 $
or,$\small [E(X^2)]^{\frac{1}{2}} \geq E(X)$
Q.Show that mean deviation can not exceed standard deviation.
Ans:
We know that $\small V(X) \geq 0$
or,$\small E(X^2)\geq [E(X)]^2 $
let $\small X=Y-\bar Y$
or, $\small E(Y-\bar Y)^2\geq [E(Y-\bar Y)]^2$
or,$\small \sqrt {E(Y-\bar Y)^2} \geq |E(Y-\bar Y)| \geq E|(Y-\bar Y)|$
or, Standard deviation $\small \geq $ Mean deviation
Q. If a person gets Rs. 2x+5, where x denotes the number appearing when a balanced die is rolled once, how much money can he except in the long run per game.
Ans:
The outcomes from a throwing of a balanced die is: {1,2,3,4,5,6}
Excepted money he can earn: $\small 7\times \frac{1}{6}+9\times \frac{1}{6}+...+17\times \frac{1}{6}=\frac{72}{6}$
Nice
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