MCQs on Expectation
The average value of a random phenomenon is also termed as its mathematical expectation or expected value.
1. $E(1/X)$=${1/E(X)}$ Is it true?
a.Yes
b.No
Ans: No
2.$V(X+b)$=$?$
a.$V(X)$
b.$V(X)+V(b)$
c.$V(X)+b^2$
Ans: $V(X)$
3.If $X$ and $Y$ are independent then $Cov(X,Y)$ =?
a.$0$b.$E(XY)$
c.$E(X)E(Y)$
Ans: $0$
4.A coin is tossed until a head appears. What is the expectation of the number of tosses?
a.2b.4
c.6
d.None
Ans: 2
5.Let the mean of the X is $\mu $, and the variance is $\sigma^2$. Then $E(X-b)^2$ will be min if
a.$b$= $\mu $
b.$b$=$\sigma^2$
b.$b$=$\sigma^2$
c.$b$=$X^2$
d.None
d.None
Ans: $b$= $\mu $
6. Find the correct solution$V(X)$=?
a.$E[V(X|Y]+V[E(X|Y)]$
b.$E[V(Y|X]+V[E(X|Y)]$
b.$E[V(Y|X]+V[E(X|Y)]$
c.$E[V(Y|X]+V[E(Y|X)]$
d.None
d.None
Ans: $E[V(X|Y]+V[E(X|Y)]$
7. If $Var(X)$=$1$ then $Var(2X\pm 3)$=?
a.5
b.13
c.4
b.13
c.4
Ans: 4
8.$Cov(aX+bY.,bX+aY)$=?
a.$abVar(X+Y)$
b.$abCov(X+Y)$
b.$abCov(X+Y)$
c.$aCov(X+Y)$
Ans: $abVar(X+Y)$
9. Let $f(x,y)$=$8xy$, $0<x<y1$; 0 elsewhere. What will be the value of $E(Y|X=x)$=?
a. $\frac{2}{3} \frac{1+x+x^2}{1+x}$
b. $\frac{2}{3} \frac{x(1+x+x^2)}{1+x}$
c. $\frac{1+x^2}{2}$
d. None
b. $\frac{2}{3} \frac{x(1+x+x^2)}{1+x}$
c. $\frac{1+x^2}{2}$
d. None
Ans: $\frac{2}{3} \frac{1+x+x^2}{1+x}$
10. If $P(X\leqslant Y)=1$
a. $E(X) \leqslant E(Y)$
b. $E(X) = E(Y)$
c. $E(X)\geq E(Y)$
Do it yourself:
1.If $f(x,y)=4xy$ for $0<x<1$, $0<y<1$, then find $E(Y|x)=?$, $V(Y|x)=?$.
2. Let $X$ and $Y$ is two iid variable and the mean of $X$ and $Y$ is 10,20. and the var are 2,3. Find the var of $3X+4Y$.
3.A coin is tossed four times. Let $X$ denote the number of times a head is followed immediately by a tail. Find the distribution , mean and the variance.
4.
a. Find the values of $V(X)$, $V(Y)$
b. Find the $V(Y|X=-1)$.
b. Find the $V(Y|X=-1)$.
5. What is the relation between cdf and Expectation?