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

Essay Question▾

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 freedo

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Question number: 2

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

Essay Question▾

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

Essay Question▾

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 dis

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Question number: 4

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2013

Essay Question▾

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 functi

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Question number: 5

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

Essay Question▾

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

Essay Question▾

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

Essay Question▾

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

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