a) Define continuous random variable. Also show that $E(X_{1}X_{2}...X_{n}) = E(X_{1})E(X_{2})...E(X_{n})$ (Unit 2)

b) Define non-central and central moments. Also, show that $V(2X+3)=4V(X)$ (Unit 1)

a) Show that Poisson distribution is a limiting case of binomial distribution under the case $n \rightarrow \infty$, $p \rightarrow 0$ and $np=m$. (Unit 1)

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

a) If $X \sim N(\mu, \sigma^2)$, then show that $\varphi_{x}(t) = e^{i\mu t - \frac{1}{2}t^2\sigma^2}$ (Unit 2)

b) A large number of measurement is normally distributed with a mean 65.5 cm and S.D. of 6.2 cm. Find the percentage of measurement that fall between 54.8 cm and 68.8 cm. (Unit 2)

a) Write down merits and demerits of measure of central tendency. Find the median for the following data; (Unit 4)

b) Show that sum of two independent random variables follows normal distribution. (Unit 1)

a) Using method of Least squares, find the curve $y=ax+ax^{2}$ that best fit the following data: (Unit 5)

b) Out of 8000 graduates in a town, 800 are females, out of 1600 graduate employees, 120 are females, Use $\chi^2$ test to determine if any distinction is made in appointment on the basis of sex. The value of $\chi^2$ for 1 degree of freedom at 5% level is 3.841. (Unit 6)

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

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

a) What do you mean by measure of kurtosis? Obtain $\beta_{1}$ and $\beta_{2}$ for the following function: (Unit 4)

$f(x) = y_{0}x(2-x)$; $0 \le x \le 2$

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

a) Find the rank correlation coefficient to the following data: (Unit 4)

b) Define binomial distribution and derive its moment generating function. Hence, obtain its mean and variance. (Unit 4)