A random variable x has the following probability function

(i) Determine k
(ii) Evaluate P(x < 6) and P(0 $\le$ x $\le$ 4).
(iii) If P(x $\le$ k) > $(1/2)$ find the minimum value of k.
(iv) Determine the distribution function of x.
(v) Mean

a) The density function of a random variable X is f(x)=$e^{-x}$ when x $\ge$ 0. Find E(X), E($X^2$) and variance of X. (Unit 2)

b) Suppose a continuous random variable X has the probability density function f(x)= k(1-$x^2$) for 0 < x < 1, and f(x)=0 otherwise.
Find (i) k (ii) Mean (iii) Variance. (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 3)

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

b) 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)

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 $\theta$ and obtain its mean, variance and moment generating function. (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
(i) at least one (ii) at most one. (Unit 4)

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

a) Find Karl Pearson's coefficient of correlation from the following data - (Unit 4)

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