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

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Appeared in Year: 2011

Essay Question▾

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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 standar

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Question number: 2

» Probability » Distribution Function » Standard Probability Distributions

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Appeared in Year: 2010

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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

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Question number: 3

» Probability » Distribution Function » Standard Probability Distributions

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Appeared in Year: 2015

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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

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Question number: 4

» Probability » Distribution Function » Standard Probability Distributions

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Appeared in Year: 2013

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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

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Question number: 5

» Probability » Distribution Function » Standard Probability Distributions

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Appeared in Year: 2015

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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

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Appeared in Year: 2010

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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

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Question number: 7

» Probability » Distribution Function » Standard Probability Distributions

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Appeared in Year: 2011

Essay Question▾

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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

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