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#### Random Variable and Mathematical Expectation Book Back Questions

12th Standard EM

Reg.No. :
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Time : 00:45:00 Hrs
Total Marks : 30
5 x 1 = 5
1. Value which is obtained by multiplying possible values of random variable with probability of occurrence and is equal to weighted average is called

(a)

Discrete value

(b)

Weighted value

(c)

Expected value

(d)

Cumulative value

2. Probability which explains x is equal to or less than particular value is classified as

(a)

discrete probability

(b)

cumulative probability

(c)

marginal probability

(d)

continuous probability

3. If X is a discrete random variable and p x ( ) is the probability of X , then the expected value of this random variable is equal to

(a)

$\sum { f(x) }$

(b)

$\sum { [x+f(x)] }$

(c)

$\sum { f(x)+x }$

(d)

$\sum { xp(x) }$

4. Which of the following is not possible in probability distribution?

(a)

$\sum { p(x)\ge 0 }$

(b)

$\sum { p(x)=1 }$

(c)

$\sum { xp(x)=2 }$

(d)

$p(x)=-0.5$

5. A discrete probability distribution may be represented by

(a)

table

(b)

graph

(c)

mathematical equation

(d)

all of these

6. 3 x 2 = 6
7. Construct cumulative distribution function for the given probability distribution.

 X 0 1 2 3 P(X=x) 0.3 0.2 0.4 0.1
8. The discrete random variable X has the probability function

 X 1 2 3 4 P(X=x) k 2k 3k 4k

Show that k =0.1.

9. Two coins are tossed simultaneously. Getting a head is termed as success. Find the probability distribution of the number of successes.

10. 3 x 3 = 9
11. If $p(x)\begin{cases} \underline { x } , \\ 20 \\ 0, \end{cases}x=$0,1,2,3,4,5
Find (i) P(X<3) and (ii) P(2$\le$4)

12. A coin is tossed thrice. Let Xbe the number of observed heads. Find the cumulative distribution function of X.

13. A continuous random variable X has p.d.f
f(x)=5x4,0$\le$x$\le$
Find a1 and a2 such that i) P[X$\le$a1]=P[X>a1]   ii) P[X>a2]=0.05

14. 2 x 5 = 10
15. Determine the mean and variance of a discrete random variable, given its distribution as follows.

 X=x 1 2 3 4 5 6 Fx(x) $\frac{1}{6}$ $\frac{2}{6}$ $\frac{3}{6}$ $\frac{4}{6}$ $\frac{5}{6}$ 1
16. A commuter train arrives punctually at a station every 25 minutes. Each morning, a commuter leaves his house and casually walks to the train station. Let X denote the amount of time, in minutes, that commuter waits for the train from the time he reaches the train station. It is known that the probability density function of X is
f(x)={\(\frac { 1 }{ 25 } ,\quad for0
Obtain and interpret the expected value of the random variable X.