# Probability-Distribution Function (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 7 of 7

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## Question number: 1

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

### Describe in Detail

Show that the square of the one sample t-statistic has the F-distribution. What are its degrees of freedom?

### Explanation

The t-statistic is defined as the ratio of a standard normal variable X~N (0, 1) and the square root of where Y~ and n is the degree of freedom.

Then we show the square of t-statistic follows F-distribution.

We known that X is standard normal distribution, then X ^{2} follows a chi-square distribution with degree of freedom is 1. Let assume …

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## Question number: 2

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

### Describe in Detail

Let (X, Y) be jointly distributed with p. d. f.

Find marginal probability density function of X and Y.

### Explanation

The marginal probability density function of X is

The range is 0 < x < 1

The marginal probability density function of Y is

The range is 0 < y < 1

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## Question number: 3

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

### Describe in Detail

Let Y _{1} denote the first order statistic in a random sample of size n from a distribution that has the pdf

Obtain the distribution of Z _{n} =n (Y _{1} -θ).

### Explanation

To find the distribution of order statistic, the pdf is

Here denote the first order statistic that is

The cdf of x is

The pdf of Y _{1} is

Then, the distribution of Z _{n} =n (Y _{1} -θ) is

The pdf of Z _{n} is

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## Question number: 4

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2013

### Describe in Detail

Find the distribution of the ratio of two iid random variables with density function:

### Explanation

Let consider x and y are two iid random variable with density function is same. Then find the distribution of their ratio that is x/y. The joint pdf of x and y is

To find this we use Jacobian transformation technique, assume

X/Y = u, Y = v

X = uv and Y = v

So, the joint density function of u and v is

The limit of v is 0 to and lim…

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## Question number: 5

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

### Describe in Detail

Let X have pdf

Obtain the cdf of Y = X ^{2}.

### Explanation

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## Question number: 6

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

### Describe in Detail

Let k > 0 be a constant, and

Obtain P (X > 0.3).

### Explanation

We first find the k value for which the f (x) is purely probability density function that is

So, density is

Then obtain P (X > 0.3)

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## Question number: 7

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

### Describe in Detail

Show that is a probability density function for an appropriate value of a. Upto what order do the moments of this p. d. f. exist?

### Explanation

To find the value of a, we known that the probability density function under the range of x is equal to one.

To find order which is exist for this p. d. f. is

Upto moments is exist is k-2 < 0 = k < 2

That is only we find the first moment for k = 1 which is mean

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