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 Equation where Y~ Equation and n is the degree of freedom.

Equation

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

Equation

We known that X is standard normal distribution,… (104 more words) …

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.

Equation

Find marginal probability density function of X and Y.

Explanation

The marginal probability density function of X is

Equation

Equation

Equation

The range is 0 < x < 1

The marginal probability density function of Y is

Equation

Equation

Equation

The range is 0 < y < 1

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

Equation

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

Explanation

To find the distribution of order statistic, the pdf is

Equation

Here denote the first order statistic that is

Equation

The cdf of x is

Equation

Equation

The pdf of Y 1 is

Equation

Equation

Equation

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

Equation

Equation

Equation<span class="more">… (10 more words) …</span>

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:

Equation

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

Equation

To find this we use Jacobian transformation technique, assume

X/Y = u, Y = v

X… (54 more words) …

Question number: 5

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Let X have pdf

Equation

Obtain the cdf of Y = X 2.

Explanation

Equation

Equation

Equation

Equation

Equation

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

Equation

Obtain P (X > 0.3).

Explanation

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

Equation

Equation

Equation

Equation

Equation

So, density is

Equation

Then obtain P (X > 0.3)

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Question number: 7

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that Equation 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.

Equation

Equation

Equation

Equation

Equation

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

Equation

Equation

Upto moments is exist is k-2 < 0 =… (19 more words) …

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