INTRODUCTORY PROBABILITY-2A — DU B.Sc (hons) Statistics Previous Year Question Papers

3. (a) A variable X is distributed at random between the value 0 and 4 and its probability density function is given by Find the value of k, the mean and standard deviation of the distribution. (b) What is a random variable? Explain briefly. Also, distinguish between discrete and continuous random variables with examples. (8,7) 4. (a) Given the following table: Compute E(X), E(X2) and E(X + 1)2. (b) Find the moment generating function of the random variable X having the p.d.f. lex forx > 0 / (x) = 1 0, elsewhere and use it to find an expression for 4'p: (8,7) 5. (a) A manufacturer, who produces medicine bottles, find that 0.1% of the bottles are defective. The bottles are packed in boxes containing 500 bottles. A drug manufacturer buys 100 boxes from the producer of bottles. Find how many boxes will contain (1) no defective and (il) at least two defectives [Given e 0.5 = 0.6065] (b) Obtain the moment generating function of Binomial distribution. Derive the result that the sum of two independent binomial variate is a binomial variate stating clearly the assumptions made. (8,7) 6. (a) Define Geometric distribution with example. Also, find its mean and variance. (b) If X and Y are independent Poisson variates with parameters 2 and u respectively such that P(X = 1) = P (X = 2) and P(Y = 2) = P(Y = 3) Find the variance of X - 2Y. (8, 7) 7. (a) Define the Hypergeometric Distribution. Find its mean and variance. (b) Mention five chief characteristic of the normal probability distribution. Also write its probability density function. (8,7) 8. (a) Assuming the mean height of students is an exactly normal distribution be 68.22 inches and variance of 10.8 inches squares. How many students in a college of 1000 students would you expect to be: (i) over six feet tall (ii) below 65 inches? (b) Define Beta Distribution of first kind. Also, obtain its mean and variance. (8,7)