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,… (104 more words) …
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
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
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… (54 more words) …
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
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)
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 =… (19 more words) …