MCQs on Expectation

 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.2
b.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.$b$=$\sigma^2$

c.$b$=$X^2$
d.None

Ans:  $b$= $\mu$

 

6. Find the correct solution$V(X)$=?

 

a.$E[V(X|Y]+V[E(X|Y)]$
b.$E[V(Y|X]+V[E(X|Y)]$

c.$E[V(Y|X]+V[E(Y|X)]$
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

 

Ans: 4

 

8.$Cov(aX+bY.,bX+aY)$=?

 

a.$abVar(X+Y)$
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

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. 

$X$ $Y$	-1	0	1	Total
-1	0	.1	.1	.2
0	.2	.2	.2	.6
1	0	.1	.1	.2
Total	.2	.4	.4	1.0

 

a. Find the values of $V(X)$, $V(Y)$
b. Find the $V(Y|X=-1)$.  

5. What is the relation between cdf and Expectation?