Probability-Distribution Function (ISS 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▾

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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 Y/n where Y~ χn2 and n is the degree of freedom.

t=XYn

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

t… (430 more words) …

Question number: 2

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

Essay Question▾

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

f(x,y)={2;0<x<y<10;otherwise

Find marginal probability density function of X and Y.

Explanation

The marginal probability density function of X is

g(x)=f(x,y)dy

=x12dy

=2[1x]

The range is 0 < x < 1

The marginal probability… (40 more words) …

Question number: 3

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

Essay Question▾

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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

f(x)={e(xθ),θ<x<0,otherwise

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

Explanation

To find the distribution of order statistic, the pdf is

fx(j)(x)=n!(j1)!(nj)!f(x)[F(x)]j1[1F… (290 more words) …

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:

f(x)=ex,0<x<0,elsewhere

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

f(x,y)=e(x+y)

To find this… (190 more words) …

Question number: 5

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

Essay Question▾

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Let X have pdf

f(x)={13;1<x<20;otherwise

Obtain the cdf of Y = X 2.

Explanation

FY(y)=P(Yy)=P(X2y)

=P(yXy)

=yyf(x)dx

=13yydx… (4 more words) …

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

f(x)={kx(1x);0<x<10;otherwise

Obtain P (X > 0.3).

Explanation

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

01f(x)dx=1

k01x(1x)dx=1

k[x22x… (135 more words) …

Question number: 7

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that f(x)=ax3,a<x,a>0 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.

af(x)dx=1

aa1x3dx=1

a[12x… (114 more words) …

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