a) If X is a continuous random variable and $k$ is a constant. Then prove that $Var(X+k)=Var(X)$ and $Var(kX)=k^{2}Var(X).$ (Unit 1)

b) State and Prove Chebyshev's inequality. (Unit 1)

a) A random variable X is defined as the sum of the numbers on the faces when two dice are thrown. Find the mean of X. (Unit 1)

b) Explain Exponential and Gamma densities with properties. (Unit 2)

The probability density function of a random variable X is (Unit 2)

$$f(x) = \begin{cases} \frac{\sin x}{2}, & \text{for } 0 \le x \le \pi \\ 0, & \text{elsewhere} \end{cases}$$

Find the mean, mode and median of a distribution and also find the probability between 0 and $\frac{\pi}{2}$.

a) Define Bivariate distribution and explain their properties. (Unit 3)

b) If first box contains 2 black, 3 red, 1 white balls; Second box contains 1 black, 1 red, 2 white balls; Third box contains 5 black, 3 red, 4 white balls. Of these a box is selected at random. From it a red ball is randomly drawn. If the ball is red, find the probability that it is from second box. (Unit 3)

a) The values of $\mu_{1},$ $\mu_{2}$, $\mu_{3}$ and $\mu_{4}$ are 0, 9.2, 3.6 and 1.22 respectively. Find out the Skewness and Kurtosis of the distribution. (Unit 4)

b) The mean and standard deviation of a normal variable are 8 and 4 respectively. Find: $P(5\le X\le10)$ and $P(X\ge5)$. (Unit 2)

a) Average number of accidents on any day on a national highway is 1.6. Determine the probability that the number of accidents are: (Unit 4)

i) at least one

ii) at most one

b) Define correlation. Explain types of correlation and methods of studying correlation. (Unit 4)

a) Fit a Straight line for the following data. (Unit 5)

b) A sample of 400 items is taken from a population whose standard deviation is 10. The mean of the sample is 40. Test whether the sample has come from a population with mean 38. Also calculate 95% confidence interval for the population. (Unit 5)

a) Define Chi-Square test of goodness of fit. Explain the conditions for Chi-Square test. (Unit 6)

b) Calculate coefficient of correlation from the following data. (Unit 4)