Probability-Standard Probability Distributions (ISS 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

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

where , 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).

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… (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.

First find the cdf of Y

Let

The upper limit is c= , lower limit is

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

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: 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

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… (9 more words) …

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