State and prove the addition rule for two events. Extend it to more than two events.

A card is drawn from a pack of 52 cards. Find the probability of getting a king or a heart or a red card.

Three technicians Y and Z service 30% and 50% breakdowns, respectively, occurring on an automated production line. The technician X makes an incomplete repair 1 time in 20, Y makes an incomplete repair 1 time in 10, and Z makes an incomplete repair 1 time in 15. For the next breakdown a repair made was found to be incomplete. Find the probability that this repair was made by Z.

What is n random variable? What are its two types? Explain with suitable examples. Define the distribution function of a random variable and state its important properties.

A consignment of 10 similar PCs contains 4 defective PCs. If an institution makes a random purchase of 3 PCs from this consignment, find the probability distribution for the number of defective PCs purchased and compute the distribution function. Also draw its graph.

Suppose that 5 out of 20 new buildings in a city violate the building code. What is the probability that a building inspector, who randomly selects 10 of the new buildings, will catch exactly 2 of the new buildings that violate the code?

Find the m.g.f. of a normal variate with mean 0 and variance 1. Show that all its odd order moments are zero.

A machine automatically packs a chemical fertilizer in polythene packets. It is observed that 10% of the packets weigh less than 2.42 kg, while 15% of the packets weigh more than 2.50 kg. Assuming the weight of the packet is normally distributed, find the mean and the variance of the packet.

A system contains a certain type of component whose lifetime X is exponentially distributed with mean of 5 years. If 8 such components are installed in different systems, then find the probability that at least 3 are still working at the end of 7 years.

Let X1 and X2 be two independent random variables each distributed uniformly in the interval where a 0 is a constant. Find the joint distribution of X1 X2 and X1 -X2·

The mean and variance of a population are m and o^2 respectively. Find the mean and variance of the sample mean of a sample of size n.

Two random samples gave the following results:

Sample Size(n) Sample mean

I 10 15 90
II 12 14 108

Test whether the sample comes from the same normal population at level of significance.

Let X1, X2, ... Xn be a random sample from a population with mean m and variance Show that

EiXi to is a consistent estimator of m.

The following are 10 measurements on some characteristic measured by the same instrument by two technicians A and B. Can we say that B is more consistent than A at level of significance?

13 15 7 15 5 12 9 3 20 11
12 7 2 8 6 9 5 7 6 8

If 41 of 120 tyres of brand A failed to last 20,000 miles, while the corresponding figures for brand B and brand C of tyres are 27 of 80, and 22 of 100 respectively, test at level of significance whether the three brands of tyres differ in quality.