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1. Which of the following statements are true or false? Give reason in support of your answer.

a) When two dice are thrown simultaneously then total number of sample points in the sample space will be 12 .

b) Expected value of a continuous random variable \(X\) is defined as \(E(X)=\int_{-\infty}^{x} x f(x) d x\).

c) If \(X\) and \(Y\) are independent random variable then \(\mathrm{V}(\mathrm{X}-\mathrm{Y})=\mathrm{V}(\mathrm{X})-\mathrm{V}(\mathrm{Y})\).

d) If \(X \sim B(4,3)\) then variance of \(X\) is 12 .

e) If probability density function of a normally distributed random variable \(\mathrm{X}\) is

\[
f(x)=\frac{1}{6 \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-46}{6}\right)^{2}}, \quad-\infty<x<\infty
\]

then variance of \(\mathrm{X}\) is 36 .

2. An insurance company selected 6000 drivers from a city at random in order to find a relationship between age and accidents. The following table shows the results to these 6000 drivers.

If a driver from the city is selected at random, find the probability of the following events:

a) Age lying between \(18-25\) and meet 3 accidents

b) Age lying between \(18-40\) and meet 1 accident

c) Age more than 25 years and meet at most one accident

d) Having no accident in the year

e) Age lying between \(18-40\) and meet at least 3 accidents

3. Determine the constant \(\mathrm{k}\) such that the function \(\mathrm{f}(\mathrm{x})=\mathrm{kx}^{2}(1-\mathrm{x})^{5}, 0<\mathrm{x}<1\) is a beta distribution of first kind. Also, find its mean and variance.

4. An insurance company insured 2000 scooter drivers, 3000 car drivers and 5000 truck drivers. The probabilities that scooter, car and truck drivers meet an accident are 0.02, 0.04, and \(0.25\) respectively. One of the insured persons meets with an accident. What is the probability that he is a

a) Scooter driver

b) Car driver

5. The following table represents the joint probability distribution of the discrete random variable \((\mathrm{X}, \mathrm{Y})\) :

Find

a) The marginal distributions.

b) The conditional distribution of \(Y\) given \(X=2\)

6. a) A rain coat dealer can earn Rs 800 per day during a rainy day. If it is a dry day, he can lose Rs 150 per day. What is his expectation, if the probability of rain is \(0.6 ?\)

b) A player tosses two unbiased coins. He wins Rs. 10 if 2 heads appear, Rs. 5 if one head appears and Rs 1 if no head appears. Find the expected value of the amount won by him.

7. a) (i) Let \(\mathrm{X}\) and \(\mathrm{Y}\) be two independent random variables such that \(\mathrm{X} \sim \mathrm{B}(5,0.06)\) and \(\mathrm{Y} \sim \mathrm{B}(4,0.6)\). Find \(\mathrm{P}[\mathrm{X}+\mathrm{Y}>1]\)

(ii) Comment on the statement: “The mean of a binomial distribution is 4 and variance 5”.

b) If the probability that an individual suffers a bad reaction from an injection of a given serum is \(0.002\), determine the probability that out of 400 individuals

(i) exactly 2

(ii) more than 3

(iii) at least one individuals suffer from bad reaction.

8. a) A die is rolled. If the outcome is a number greater than 2 , what is the probability that it is an odd prime number?

b) A person is known to hit the target in 3 out of 4 shots whereas another person is known to hit 2 out of 5 shots. Find the probability that the target being hit when they both try.

c) Events A, B, C are mutually exclusive and exhaustive. If odds against A are \(4: 1\) and against \(\mathrm{B}\) are \(3: 2\). Find the odds against event \(\mathrm{C}\).