# Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 53 - 59 of 72

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## Question number: 53

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2015

### Describe in Detail

Let X follow log-normal with parameters µ and σ ^{2}. Find the distribution of Y = aX ^{b}, a > 0, -∞ < b < ∞

### Explanation

If X follow log-normal with parameters µ and σ ^{2}, then Z = logX follow normal distribution.

First find the cdf of Y

Let

The upper limit is c= , lower limit is

Putting the value of c

## Question number: 54

» Probability » Characteristic Function

Appeared in Year: 2009

### Describe in Detail

Find the density, if its characteristic function is

### Explanation

Rewrite the characteristic function Φ (t)

The density function of X is given by Fourier inversion theorem using the characteristic function,

The general form is

Putting the characteristic function

## Question number: 55

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2014

### Describe in Detail

If f _{x} (x) be the probability density function of a lognormal distribution, show that

Where and upper limit is and φ (z) is the distribution function of the standard normal distribution. Hence find E (X) and V (X).

### Explanation

Given that f _{X} (x) has a lognormal distribution, the probability density function is

Then

………. (1)

Let assume that =z, then differentiate

The limit is also change, the lower limit is and upper limit is

Equation (1) can be written as

## Question number: 56

» Probability » Conditional Probability

Appeared in Year: 2009

### Describe in Detail

(i) Let X be a random variable such that P [X < 0] = 0 and E [x] exist. Show that P (X ≤ 2E [x] ) ≥ l/2

(ii) Let E [X] = 0 and E [X ^{2}] be finite. Show that P (X ^{2} < 9E [X ^{2}] ) > 8/9

### Explanation

(i) Using Markov inequality, for any random variable and constant a > 0

Here a = 2E (X)

or

(ii) Using Chebyshev inequality, Let X have mean E (X) =µand Var (X) =σ ^{2}, then for any a > 0

In this

## Question number: 57

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2009

### Describe in Detail

Suppose that the random variable X has a normal distribution with mean µ and variance σ ^{2}. Let φ be the distribution function of a standard normal variate. Find the density of φ (X-µ/σ). Also find E [φ (X-µ/σ) ].

### Explanation

X has a normal distribution with mean µ and variance σ ^{2}

Let

Let t=

σdt = dx, the upper limit is t = z and the lower limit is same

The density function is

The mean is

The integral

## Question number: 58

» Probability » Conditional Probability

Appeared in Year: 2011

### Describe in Detail

Let (X, Y) have the uniform distribution over the range 0 < y· < x < 1. Obtain the conditional mean and variance of X given Y = y.

### Explanation

The joint probability density function of (X, Y) is

The marginal distribution of X is

The conditional distribution of X given Y = y

The conditional mean is

The conditional variance is

## Question number: 59

» 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*