Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 26 - 31 of 72

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Question number: 26

» Probability » Characteristic Function

Appeared in Year: 2013

Essay Question▾

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Find the density function of X whose characteristic function Φ (t) is given by:

ϕ(t)={1 t , t 10, t >1

Explanation

Rewrite the characteristic function Φ (t)

ϕ(t)={1t,0<t11+t,1<t<00,t>1ort<1

ϕ(t)={1… (236 more words) …

Question number: 27

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2010

Essay Question▾

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Let X be a random variable with a continuous distribution function F. Show that F (X) has the uniform distribution on (0, 1).

Explanation

Let U = F (x), then the distribution function of G of U is given by

G(u)=P[Uu]=P[F(X)u]=P[XF1(u)]

The… (93 more words) …

Question number: 28

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2012

Essay Question▾

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For the Cauchy distribution given by

f(x)=kσ2+(xμ)2,<x<

where k is a constant to be suitably chosen, derive the expression for the distribution function. Hence obtain a measure of central tendency and a measure of dispersion. What are the points of inflexion of the distribution?

Explanation

We choose k as constant that gives the integral over the range x for the density function is equal to one.

kσ2+(xμ)2dx=1

kσ[tan1(xμσ… (381 more words) …

Question number: 29

» Probability » Sample Space

Appeared in Year: 2011

Essay Question▾

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A fair die is thrown until a 6 appears. Specify the sample space. What is the probability that it must be thrown at least 3 times?

Explanation

A fair die content the value {1, 2, 3, 4, 5, 6}

So, the probability of getting 6 is p = 1/6, then probability of getting other than 6 is q = 5/6

If we throw a die, the six is appear. Then the probability is p and the experiment… (162 more words) …

Question number: 30

» Probability » Conditional Probability

Appeared in Year: 2011

Essay Question▾

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Consider the following bivariate p. m. f. of (X, Y):

p (0, 10) = p (0, 20) = 2/18;

p (l, 10) = p (l, 30) = 3/18;

p (1, 20) = p (2, 30) = 4/18;

Obtain the conditional mass functions p (y lx = 2), and p (y lx = 1).

Explanation

The joint probability is given

-

x

Total

-

0

1

2

Y

10

2/18

3/18

0

5/18

20

2/18

4/18

0

6/18

30

0

3/18

4/18

7/18

Total

4/18

10/18

4/18

1

First find the marginal probability of x at x = 0, 1, 2

p(x)… (136 more words) …

Question number: 31

» 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.

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) …

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