MCQs on random variables and distribution functions

 Random variables and Distribution functions

 

This chapter is consist of 

A/Distribution function
B/Discrete Random Variable 

C/Continuous Random Variable
D/Two dimensional random variable

 

1. If f is the distribution function of the r.v X then $F(b)-F(a)-P(X=b)+P(X=a)$ =?

 

a.$P( a \leqslant X <b)$
b.$P( a \leqslant X \leqslant b)$

c.none 

Ans: $P( a \leqslant X <b)$ 

 

2.

$x$	0	1	2	3	4	5	6	7
$f(x)$	0	$k$	$2k$	$2k$	$3k$	$k^{2}$	$2k^{2}$	$7k^{2}+k$

What is the value of $k$?

 

a.-1

b.2

c.3
d.-2

Ans: -1 

3. Consider a pdf $f(x)$ = $6x(1-x)$, ), $0 \leqslant X \leqslant1$

What will be the value of b, if $P(X<b)$ = $P(X>b)$?

 

a.0.5
b.0.7
c.0.4
d.0.2

 

Ans: 0.5

 

4. Let G the geometric mean of the distribution and $dF$=$6(2-x)(x-1)$, $1 \leqslant X \leqslant2$

Choose the correct one?

 

a.$6 log(16G)=19$
b.$16 log(6G)=19$
c.$6 log(19G)=16$
d.None

Ans: $6 log(16G)=19$

 

5.What is the mean deviation of the distribution  $f(x)$ = $\frac{3+2x}{18}$ for $2 \leqslant x \leqslant4$,$0$ otherwise?

 

a.0.49
b.0.36
c.0.54
d.0.25

Ans:  0.49

 

6.

$x$	$0\leqslant x <1$	$1\leqslant x <2$	$2\leqslant x <3$	$otherwise$
$f(x)$	$kx$	$k$	$-kx +3k$	$0$

What will be the value of $P(X>1.5)$=?

 

a.$\frac{1}{2}$
b.$\frac{3}{4}$
c.$\frac{1}{4}$

d.None 

 Ans: $\frac{1}{2}$

 

7.

$x$	$x <0$	$0\leqslant x <2$	$2\leqslant x <4$	$x\geqslant 4$
$F(x)$	$0$	$\frac{x}{8}$	$\frac{x^2}{16}$	$1$

What will be the value of $P(X<2)$=? [ F(x) = cdf of the r.v X]

 

a.3/4
b.1/4
c.2/4
d.None

 

Ans: 1/4

 

8. MARGINAL DISTRIBUTION OF X 

$x$	$1$	$2$	$3$	$4$
$P(X=x$	$10/36$	$9/36$	$8/36$	$9/36$

Find the value of $P(X=1|Y=1)$=?

 

a.4/11
b.1/11
c.5/11
d.3/11
e.10/11

 

Ans: 4/11

9.

$x$	$0 \leqslant x <2$, $2\leqslant x <2$	$otherwise$
$f(x)$	$\frac{1}{8} (6-x-y)$	$0$

What will be the value of $P(X<1 \cap Y<3)$ 

 

a.3/8
b.4/8
c.5/8
d.None

 

Ans: 3/8

 

10.

$x,y$	$\left | x \right |<1$,$\left | y \right |<1$	$otherwise$
$f_{XY}(x,y)$	$\frac{1}{4} (1+xy)$	$0$

 

If $X$ and $Y$ are not independent by $X^2$,$Y^2$ are independent.Is it true?

 

a.Yes
b.No.

Ans: Yes

 

Do it yourself:  

 

1.The joint probability distribution for two random variables $X$ and $Y$ is given by : $P(X=0,Y=1)=1/3$, $P(X=1,Y=-1)=1$ and $P(X=1,Y=1)=1/3$. What is the conditional distribution of $X$ given $Y=1$? 

 

2.If,$P(0)$= $3c^3$,$P(1)$=$4c-10c^2$,$P(2)$=$5c-1$ for some $c$>$0$.a.Determine the value of the $c$?b.Compute the probabilities of $P(X,2)$ and $P(1<X<2)$. 

3.If $f(x)$=$A+Bx$, $0\leqslant x \leqslant 1$, and the mean is $\frac{1}{2}$.Find the value of the A and B.

4. .If $f(x)$=$\frac{1}{2}(1+x)$, $-1\leqslant x \leqslant 1$, then find the Skewness and Kurtosis.

5.

$X$	$0$	$1$	$2$	$3$
$p(x)$	$0.1$	$0.3$	$0.5$	$0.1$

Let, Y= $X^2$+$2X$, Find the probability function and the mean,variance of Y.

6.

$(x,y)$	$0 \leqslant y<x<\infty$	$otherwise$
$f(x,y)$	$ke^(x+y)$	$0$

 

a.Determine $k$ ?
b.Find the conditional distribution of $f(x|y)$? 

c.Compute $P(Y \geqslant 3)$?
d.Examine if $X$ and $Y$ are independent or not?

 

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