Probability-Standard Probability Distributions (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 13 - 17 of 22

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Question number: 13

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2012

Essay Question▾

Describe in Detail

Write down the probability mass function of geometric distribution. State and prove its ‘lack of memory property’. Find also the mean and the variance of the distribution.

Explanation

If there are number of trails such that the probability of success is p. So, the probability that there are x failures before the first success is

Equation

This is the probability mass function of geometric distribution

Statement:

Among the all discrete distributions, the geometric distribution has the lack of… (169 more words) …

Question number: 14

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2014

Essay Question▾

Describe in Detail

If f x (x) be the probability density function of a N (µ, σ 2) distribution, then show that

Equation

where Equation , Equation and φ (x) and Φ (x) are the probability density function and distribution function of the standard normal distribution respectively.

Explanation

Given that f X (x) be the probability density function follows N (µ, σ 2).

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… (15 more words) …

Question number: 15

» 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: 16

» 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: 17

» 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) …

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