Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 64 - 71 of 72

Let X and Y are independent Poisson random variables with parameters λ and µ respectively. We proof this by moment generating function. The moment generating function of Poisson distribution is

So, the sum of X and Y moment generating function is

because X and Y are independent

‎

First find the marginal distribution of X and Y

The conditional density of X|Y is

The conditional density of Y|X is

‎

The Joint p. d. f. is

f (x, y) = e ^-y, 0 < x < y < ∞.

Let assume X + Y =U and Y = V, then X = U-V

Using Jacobian technique

The range is 0 < ∞, u ≤ v < ∞

The joint p. d. f. of U and V is

The marginal distribution of u is

Find that

The probability generating function is defined as

The property of memory less is that these distributions of “time from now to the next period” are exactly the same. The property is most easily explained in terms of “waiting times.

Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is said to have lack of memory if

We know that

For finding the conditional distribution, first find the distribution of sum of X ₁, X ₂, …, X _n.

Assume that y-k = n

Similarly, for last two terms,

Let A and B are two possible events in the sample space.

(i) Additive law of probability is

{Commutative property }

We know that

This shows that

(ii) This also show by additive law of probability

Let assume BUC = D, then