# 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

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## 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

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## 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

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## 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

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## Question number: 18

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2015

### Describe in Detail

Prove that for r = 1, 2, …, n

### Explanation

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