4. (a) Write the experimental conditions for binomial distribution and poisson distribution. Also prove the moment generating function for normal distribu- tion. Also let mean and variance are 2 and 4 then find P(X ≥ 20), P(X ≤ 20), P(5 < X < 10) and P(2 < X < 5).
(b) An integer is choosen at random from 200 digits. What is probability that integer is divisible by 6 or 8.
5. (a) Obtain characteristic function of standard binomial variate (X-np/√(npq)), where X is number of success obtained in n independent trial, each with constant probabilityp of success. Obtain the limit of this function as n tends to ∞. Also interpret the result.
(b) Probabilities of X, Y and Z becoming managers are 4/9, 2/9 and 1/3 respectively. Probabilities that bonus scheme will be introduced if X, Y and Z becomes managersb are 3/10 ,1/2 and 4/5 respectively.
