a) Define discrete random variables. If X and Y are any two random variables, then show that $E(X+Y) = E(X) + E(Y)$ Provided $E(X)$ and $E(Y)$ exist. (Unit 1)

b) If X is a continuous random variable and $Y = aX + b$. Prove that $E(Y) = aE(X) + b$ and $V(Y) = a^2V(X)$, where V stands for variance and a, b are constants. (Unit 2)

a) A sample of 4 items is selected at random from a box containing 12 items of which 5 are defective. Find the expected number E of defective items. (Unit 1)

b) Let $f(X)$ be the probability density function of a continuous random variable X. Then explain Mean, Median, Mode, variance and mean deviation. (Unit 2)

The probability density $f(X)$ of a continuous random variable is given by $f(x) = Ce^{-|x|}, -\infty < x < \infty$. Show that $C=\frac{1}{2}$ and find the mean and Variance of the distribution. Also find the probability that the variate lies between 0 and 4. (Unit 2)

State and prove Bayes theorem. Suppose 5 men out of 100 and 25 Women out of 10,000 are color blind. A color blind person is chosen at random. What is the probability of the person being a male. (Assuming male and female to be in equal numbers). (Unit 3)

a) The first four central moments of a distribution are 0, 2.5, 0.7 and 1.75. Calculate $\beta_1, \beta_2$. (Unit 4)

b) The mean and variance of a binomial variable X with parameters n and p are 16 and 8. Find : $P(x \ge 1)$ and $P(x > 2)$. (Unit 4)

a) If X is normally distributed with mean 2 and variance 0.1, then find $P(|x-2| \ge 0.01)$. (Unit 2)

b) Explain comparison between Correlation and Regression. (Unit 4)

a) Fit a parabola for the following data. (Unit 5)

b) In a big city 325 men out of 600 men were found to be smokers. Does this information support the conclusion that the majority of men in this city are smokers. (Unit 5)

a) The table below give the number of air craft accidents that occurred during the various days of the week. Test whether the accidents are uniformly distributed over the week. (Unit 6)

b) Show that the maximum value of rank Correlation coefficient is 1. (Unit 4)