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

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## Question number: 64

» 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: 65

» Probability » Conditional Probability

Appeared in Year: 2009

### Describe in Detail

The joint density of (X, Y) is

Find the conditional densities and E [X|Y = 1.5].

### Explanation

First find the marginal distribution of X and Y

The conditional density of X|Y is

The conditional density of Y|X is

## Question number: 66

» 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: 67

» Probability » Probability Generating Functions

Appeared in Year: 2009

### Describe in Detail

Find the generating function of X whose probability density function is

P [X = r] =pq ^{r-1}, r = 1, 2, …, 0 < p < 1, q = 1 - p

### Explanation

The probability generating function is defined as

## Question number: 68

» 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

## Question number: 69

» Probability » Expectation

Appeared in Year: 2015

### Describe in Detail

For random variables X, Y, show that

### Explanation

We know that

## Question number: 70

» Probability » Conditional Probability

Appeared in Year: 2015

### Describe in Detail

Let X _{1}, X _{2}, …, X _{n } be independent Poisson variates with E (X _{i}) =µ _{i}. Find the conditional distribution of

### Explanation

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,

Given X _{1}, X _{2}, …, X _{n } being

## Question number: 71

» Probability » Laws of Total and Compound Probability

Appeared in Year: 2011

### Describe in Detail

Verify the following identities:

(i)

(ii)

### Explanation

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

Putting the