Probability-Conditional Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 1 - 6 of 6

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Question number: 1

» Probability » Conditional Probability

Appeared in Year: 2011

Essay Question▾

Describe in Detail

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

Equation

p (0) =p… (75 more words) …

Question number: 2

» Probability » Conditional Probability

Appeared in Year: 2009

Essay Question▾

Describe in Detail

(i) Let X be a random variable such that P [X < 0] = 0 and E [x] exist. Show that P (X ≤ 2E [x] ) ≥ l/2

(ii) Let E [X] = 0 and E [X 2] be finite. Show that P (X 2 < 9E [X 2] ) > 8/9

Explanation

(i) Using Markov inequality, for any random variable and constant a > 0

Equation

Here a = 2E (X)

Equation

Equation

or

Equation

(ii) Using Chebyshev inequality, Let X have mean E (X) =µand Var (X) =σ 2, then for any a > 0

Equation

Equation

Equation

In this… (11 more words) …

Question number: 3

» Probability » Conditional Probability

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let (X, Y) have the uniform distribution over the range 0 < y· < x < 1. Obtain the conditional mean and variance of X given Y = y.

Explanation

The joint probability density function of (X, Y) is

Equation

The marginal distribution of X is

Equation

The conditional distribution of X given Y = y

Equation

The conditional mean is

Equation

Equation

The conditional variance is

Equation

Question number: 4

» Probability » Conditional Probability

Appeared in Year: 2012

Essay Question▾

Describe in Detail

You arc given the following information:

(i) In random testing, you test positive for a disease.

(ii) In 5 % of cases, the test shows positive even when the subject does not have the disease.

(iii) In the population at large, one person in 1000 has the disease. What is the conditional probability that you have the disease given that you have been tested positive, assuming that if someone has the disease, he will test positive with probability 1?

Explanation

Let X denotes the test is positive and Y denotes the person has disease. Given that 5 % of cases, the test shows positive even when the subject does not have the disease that is

Equation

where Equation denotes the person has no disease. Also given the test positive with… (41 more words) …

Question number: 5

» Probability » Conditional Probability

Appeared in Year: 2009

Essay Question▾

Describe in Detail

The joint density of (X, Y) is

Equation

Find the conditional densities and E [X|Y = 1.5].

Explanation

First find the marginal distribution of X and Y

Equation

Equation

Equation

Equation

The conditional density of X|Y is

Equation

Equation

Equation

The conditional density of Y|X is

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Question number: 6

» Probability » Conditional Probability

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Let X 1, X 2, …, X n be independent Poisson variates with E (X i) =µ i. Find the conditional distribution of

Equation

Explanation

For finding the conditional distribution, first find the distribution of sum of X 1, X 2, …, X n.

Equation

Equation

Assume that y-k = n

Equation

Similarly, for last two terms,

Equation

Equation

Equation

Equation

Given X 1, X 2, …, X n being… (67 more words) …

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