# Probability-Standard Probability Distributions (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 14 - 18 of 22

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## Question number: 14

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2014

### 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

… (140 more words) …

## Question number: 15

» 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

… (207 more words) …

## Question number: 16

» 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

… (187 more words) …

## Question number: 17

» 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

… (185 more words) …

## Question number: 18

» 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* −1)! for *r* integer.

… (108 more words) …