a) Define discrete random variable and independence of random variable. Also, show that $E(X_{1}+X_{2}+...+X_{n})=E(X_{1})+E(X_{2})+...+E(X_{n})$ (Unit 1)

b) Define expectation of random variables. Also, show that $V(aX+b)=a^{2}V(X)$ (Unit 1)

a) State Chebyshev's inequality. If $n \rightarrow \infty,$ $p \rightarrow 0$ and $np=\lambda$ then show that binomial distribution reduces to Poisson distribution. (Unit 1)

b) Define gamma distribution. If X follows exponential distribution with parameter $\theta$, then obtain its mean and variance. (Unit 2)

a) Define regression coefficient with its properties. If $X \sim N(\mu,\sigma^{2})$, then show that $M_{x}(t)=e^{\mu t+\frac{1}{2}t^{2}\sigma^{2}}$ (Unit 4)

b) Define normal distribution with its properties. Describe the methodology for difference of means for large samples. (Unit 2 & 5)

a) What do you understand by measure of central tendency? Also, write down its merits and demerits. Calculate mean and standard deviation for the observations 5, 10, 20, 25, 40, 42, 45, 48, 70, 80. (Unit 4)

b) Describe bivariate distribution. Write down the probability density function of bivariate normal distribution. (Unit 3)

a) Fit a second degree parabola to the following data: (Unit 5)

b) What do you understand by Chi-square test of goodness of fit? Write condition for applying Chi-square test. (Unit 6)

a) Define Binomial distribution and obtain its mean and variance. (Unit 4)

b) Define exponential distribution with parameter $\lambda$ and obtain its mean, variance and moment generating function. (Unit 2)

a) What do you mean by measure of skewness? Write tests of skewness. (Unit 4)

b) Define Spearman's rank correlation coefficient and obtain rank correlation coefficient for the following data: (Unit 4)

a) Find the coefficient of correlation between X and Y (Unit 4)

b) Find the regression line of y on x for the following data: (Unit 4)