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#### Probability Distributions Three Marks Questions

12th Standard EM

Reg.No. :
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Maths

Time : 00:45:00 Hrs
Total Marks : 39
13 x 3 = 39
1. Suppose two coins are tossed once. If X denotes the number of tails,
(i) write down the sample space
(ii) find the inverse image of 1
(iii) the values of the random variable and number of elements in its inverse images

2. Suppose a pair of unbiased dice is rolled once. If X denotes the total score of two dice, write down
(i) the sample space
(ii) the values taken by the random variable X,
(iii) the inverse image of 10, and
(iv) the number of elements in inverse image of X.

3. An urn contains 2 white balls and 3 red balls. A sample of 3 balls are chosen at random from the urn. If X denotes the number of red balls chosen, find the values taken by the random variable X and its number of inverse images

4. Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs.30 for each black ball selected and we lose Rs.20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.

5. Find the constant C such that the function $f(x)=\begin{cases} \begin{matrix} { Cx }^{ 2 } & 1 is a density function, and compute (i) P(1.5 < X < 3.5) (ii) P(X ≤2) (iii) P(3 < X ) . 6. 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)

7. If X is the random variable with distribution function F(x) given by,
$F(x)=\begin{cases} \begin{matrix} 0 & x<0 \end{matrix} \\ \begin{matrix} x & 0\le x<1 \end{matrix} \\ \begin{matrix} 1 & 1\le x \end{matrix} \end{cases}$
then find (i) the probability density function f (x) (ii) P(0.2≤ X ≤0.7)

8. Let X be a random variable denoting the life time of an electrical equipment having probability density function
$f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}$
Find
(i) the value of k
(ii) Distribution function
(iii) P(X < 2)
(iv) calculate the probability that X is at least for four unit of time
(v) P(X = 3)

9. Suppose that f (x) given below represents a probability mass function

 x 1 2 3 4 5 6 f(x) c2 2c2 3c2 4c2 c 2c

Find
(i) the value of c
(ii) Mean and variance.

10. 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.

11. 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

12. The mean and variance of a binomial variate X are respectively 2 and 1.5. Find
(i) P(X = 0)
(ii) P(X =1)
(iii) P(X ≥1)

13. On the average, 20% of the products manufactured by ABC Company are found to be defective. If we select 6 of these products at random and X denote the number of defective products find the probability that (i) two products are defective (ii) at most one product is defective (iii) at least two products are defective.