# Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 52 - 56 of 72

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## Question number: 52

» 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

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## Question number: 53

» 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.

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## Question number: 54

» Probability » Characteristic Function

Appeared in Year: 2009

### Describe in Detail

Find the density, if its characteristic function is

### Explanation

Rewrite the characteristic function Φ (t)

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## Question number: 55

» 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

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## Question number: 56

» Probability » Conditional Probability

Appeared in Year: 2009

### 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

Here a = 2E (X)

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