Probability (ISS 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

Essay Question▾

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.

Equation

Equation

First find the cdf of Y

Equation

Equation

Equation

Equation

Let Equation

The upper limit is c= Equation, lower limit is Equation

Equation

Equation

Putting the value of c… (8 more words) …

Question number: 54

» Probability » Characteristic Function

Appeared in Year: 2009

Essay Question▾

Describe in Detail

Find the density, if its characteristic function is

Equation

Explanation

Rewrite the characteristic function Φ (t)

Equation

Equation

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

The general form is

Equation

Putting the characteristic function

Equation

Equation

Equation

Equation

Equation

Question number: 55

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2014

Essay Question▾

Describe in Detail

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

Equation

Where Equation and upper limit is Equation 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

Equation

Then

Equation ………. (1)

Let assume that Equation =z, then differentiate Equation

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

Equation (1) can be written as

Equation

Equation<span class="more">… (60 more words) …</span>

Question number: 56

» Probability » Conditional Probability

Appeared in Year: 2009

Essay Question▾

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

Equation

Here a = 2E (X)

Equation

Equation

or

Equation

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

Equation

Equation

Equation

In this… (11 more words) …

Question number: 57

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2009

Essay Question▾

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

Equation

Let Equation

Equation

Equation

Equation

Let t= Equation

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

Equation

The density function is

Equation

The mean is

Equation

Equation

The integral… (9 more words) …

Question number: 58

» Probability » Conditional Probability

Appeared in Year: 2011

Essay Question▾

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

Equation

The marginal distribution of X is

Equation

The conditional distribution of X given Y = y

Equation

The conditional mean is

Equation

Equation

The conditional variance is

Equation

Question number: 59

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Prove that for r = 1, 2, …, n

Equation

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.

Equation

Equation

The incomplete gamma function is

Equation

Then

Equation

provided that r is an integer. Thus recall that Γ (r) = (r… (20 more words) …

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