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#### Probability Distributions Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. Let X be random variable with probability density function
$f(x)=\begin{cases} \begin{matrix} \frac { 2 }{ { x }^{ 3 } } & 0<x\ge l \end{matrix} \\ \begin{matrix} 0 & 1\le x<2l \end{matrix} \end{cases}$
Which of the following statement is correct

(a)

both mean and variance exist

(b)

mean exists but variance does not exist

(c)

both mean and variance do not exist

(d)

variance exists but Mean does uot exist

2. A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

(a)

6

(b)

4

(c)

3

(d)

2

3. Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are

(a)

i + 2n, i = 0,1,2... n

(b)

2i- n, i = 0,1,2... n

(c)

n - i, i = 0,1,2... n

(d)

2i + 2n, i = 0, 1, 2...n

4. If the function  $f(x)=\cfrac { 1 }{ 12 }$ for. a < x < b, represents a probability density function of a continuous random variable X, then which of the followingcannot be the value of a and b?

(a)

0 and 12

(b)

5 and 17

(c)

7 and 19

(d)

16 and 24

5. If P{X = 0} = 1- P{X = I}. IfE[X) = 3Var(X), then P{X = 0}.

(a)

$\cfrac { 2 }{ 3 }$

(b)

$\cfrac { 2 }{ 5 }$

(c)

$\cfrac { 1 }{ 5 }$

(d)

$\cfrac { 1 }{ 3 }$

6. 5 x 2 = 10
7. The cumulative distribution function of a discrete random variable is given by

Find
(i) the probability mass function
(ii) P(X < 1 ) and
(iii) P(X ~ 2

8. The cumulative distribution function of a discrete random variable is given by

Find
(i) the probability mass function
(ii) P(X < 3) and
(iii) P(X ~ 2).

9. If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
(i) exactly 10 will have a useful life of at least 600 hours;
(ii) at least 11 will have a useful life of at I least 600 hours;
(iii) at least 2 will not have a useful life of at : least 600 hours.

10. If μ and σ2 are the mean and variance of the discrete random variable X, and E(X + 3)=10 and E(X + 3)2 = 116, find μ and a2

11. Find the binomial distribution function for each of the following.
(i) Five fair coins are tossed once and X denotes the number of heads.
(ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

12. 5 x 3 = 15
13. Find the probability mass function and cumulative distribution function of number of girl child in families with 4 children, assuming equal probabilities for boys and girls.

14. The probability density function of X is

Find P(1.2≤X<1.8)

15. Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let X be the possible outcomes drawing red balls. Find the probability mass function and mean for X.

16. Find the mean and variance of a random variable X , whose probability density function is $f(x)=\begin{cases} \begin{matrix} { \lambda e }^{ -2x } & for\ge 0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}$

17. A multiple choice examination has ten questions, each question has four distractors with exactly one correct answer. Suppose a student answers by guessing and if X denotes the number of correct answers, find (i) binomial distribution (ii) probability that the student will get seven correct answers (iii) the probability of getting at least one correct answer

18. 4 x 5 = 20
19. The probability density function of X is given
$f(x)=\begin{cases} \begin{matrix} { Ke }^{ \frac { -x }{ 3 } } & \begin{matrix} for & x>0 \end{matrix} \end{matrix} \\ \begin{matrix} 0 & \begin{matrix} for & x\le 0 \end{matrix} \end{matrix} \end{cases}$
Find
(i) the value of k
(ii) the distribution function.
(iii) P(X <3)
(iv) P(5 ≤X)
(v) P(X ≤ 4)

20. A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station.Let X denote- the amount of time, in minutes that the student waits for the train from the time he reaches the train station. It is known  that the pdf of X is
$f(x)=\begin{cases} \begin{matrix} \frac { 1 }{ 3 } & 0<x<30 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}$
Obtain and interpret the expected value of the random variable X .

21. A six sided die is marked ‘1’ on one face, ‘2’ on two of its faces, and ‘3’ on remaining three faces. The die is rolled twice. If X denotes the total score in two throws.
(i) Find the probability mass function.
(ii) Find the cumulative distribution function.
(iii) Find P(3 ≤ X< 6) (iv) Find P(X ≥ 4) .

22. If X is the random variable with probability density functionf(x)given by,

$f(x)=\begin{cases} \begin{matrix} x-1 & 1\le x<2 \end{matrix} \\ \begin{matrix} -x+3 & 2\le x<3 \end{matrix} \\ \begin{matrix} 0 & Otherwise \end{matrix} \end{cases}$
find (i) the distribution function F (x)
(ii) P(1.5 ≤ X ≤ 2.5)