For the two events A and B show that P(A n P(A U

Suppose an assembly plant receives its voltage regulators from three different sources, 60% from B1, 30% from B2 and 10% from B3. Let and 65% of the supply received respectively from the sources B1, B2 and B3 perform as per specifications laid. If A is the event that a voltage regulator received at the. plant performs as per specification, then find

The joint density function of X and Y is given by 4xy x 0.
Test whether X and Y are independent. Also find the conditional density function of X given Y y.

Seven coins are tossed and the number of heads are noted. The experiment is repeated 128 times and the following distribution is obtained.

No.of Heads 0 1 2 3 4 5 6 7
Frequency: 7 6 19 35 30 23 7 1

Fit a binomial distribution assuming the coin to be unbiased and test the goodness of fit.

Define a Poisson variate. Find its mean and variance. Show that the sum of two independent Poisson variates is a Poisson variate.

A multiple-choice quiz has 200 questions each with 4 possible answers of which only 1 is the correct answer. What is the probability that sheer guess-work yields 25 to 30 correct answers for 80 of the 200 problems about which the student has no knowledge?

Find the correlation coefficient p between the variables defined by the p.d.f.
0 x 0 y 2

If F has F-distribution with v2) d.f., then show that 1/F follows an F distribution with d.f.

Sixteen numbers are selected independently and at random from the interval 1]. Find the probability that the mean of these sixteen numbers does not exceed 0·6.

State and prove central limit theorem.

The number of items cleared by an assembly line during a week is a random variable with mean 50 and variance 25.

What is the probability that items cleared this week will exceed 75

What can be said about the probability that this week's clearance will be between 40 and 60

Is X^2 an unbiased estimator for J.L^2, J.L being population mean? If not find its bias.

A random sample of size n is taken from a geometric distribution with parameter p. Find the moment estimator and MLE for p. Are they same?

The test runs with six models of an experimental engine showed that they operated respectively for 24, 28, 21, 23, 32 and 22 minutes with a gallon of fuel. Obtain a 99% C.I. for the average run time of engine with a gallon of fuel.

The following table gives the sample data on the number of defective castings produced by five different moulds.

Moulds: I II III IV V

Defective Castings: 14 33 21 17 25

Sample size: 100 200 180 120 150

On the basis of the data can we say that the proportion of defectives is same for different moulds? Use 0·05.