# Probability-Standard Probability Distributions (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 18 - 22 of 22

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## Question number: 18

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2015

### Describe in Detail

Prove that for r = 1, 2, …, n

### Explanation

We known that the L. H. S. is an incomplete gamma function and R. H. S. is a cumulative density function of Poisson distribution.

The incomplete gamma function is

Then

provided that *r* is an integer. Thus recall that Γ (*r*) = (*r*

## Question number: 19

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2011

### Describe in Detail

Let X have the continuous c. d. f. F (x). Define U = F (x). Show that both - log U arid -log (1 - U) are exponential random variables:

### Explanation

Let U = F (x), then the distribution function of G of U is given by

Since F is non-increasing and its continuous.

G (u) =F (F ^{-1} (u) ) implies G (u) =u

Then the p. d. f is

Since F is a distribution function takes value

## Question number: 20

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2011

### Describe in Detail

Show that the sum of two independent Poisson random variables with parameters λ and µ respectively is a Poisson random variable with parameter λ+µ.

### Explanation

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

## Question number: 21

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

### Describe in Detail

Let the joint p. d. f. of (X, Y) be f (x, y) = e ^{-y}, 0 < x < y < ∞.

Obtain the probability P (X + Y ≤ 1).

### Explanation

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

## Question number: 22

» Probability » Standard Probability Distributions » Exponential

Appeared in Year: 2009

### Describe in Detail

Explain “Memoryless property” of a distribution. Show that the exponential distribution has memoryless property.

### Explanation

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