Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 39 - 45 of 72

Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to 165 questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.

Rs. 550.00 or

Question number: 39

» Probability » Tchebycheffs Inequality

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Let X be a random variable with E [X] = 3 and E [X 2] = 13. Use Chebyshev’s inequality to obtain P [-2 < X < 8].

Explanation

Let X be a random variable with mean µ and variance σ 2. Then any k > 0, the Chebyshev’s inequality is

Equation

or Equation

σ 2 = E [X 2] - (E [X] ) 2 =4

Then, a lower bound for the probability

Equation

Using Chebyshev’s inequality

… (2 more words) …

Question number: 40

» Probability » Elements of Measure Theory

Appeared in Year: 2012

Essay Question▾

Describe in Detail

Of three independent events A, Band C, A only happens with probability ¼, B only happens with probability 1/8 and C only happens with probability 1/12. Find the probability that at least one of these three events happens.

Explanation

Given that P (A) =1/4, P (B) =1/8, P (C) =1/12

then probability that at least one event of these three events happens is

Equation

Equation

Equation

The events are independent because only one event happens

Equation

Equation

Equation

Equation

Question number: 41

» Probability » Standard Probability Distributions » Binomial

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let X 1, X 2, …, X m be i. i. d. random variables with common p. m. f.

Equation

obtain the p. m. f. of S m = X 1 + X 2 + …. + X m.

Explanation

Let X 1, X 2, …, X m i. i. d. random variables with common p. m. f. is P (X = k) which is a binomail random variables with common parameters n and p respectively. Then, the p. m. f. of S m = X 1 +

… (117 more words) …

Question number: 42

» Probability » Standard Probability Distributions » Gamma

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let

Equation

Show that f (x) is a probability density function. Obtain V (X).

Explanation

if X is a continuous random variable and f (x) is a continuous function of X, then f (x) is a probability density function if

Equation

Equation

Equation

Assume Equation but limit is same

Equation

Equation

This integral is a gamma function Equation

So,

Equation

Thus f (x) is a

… (57 more words) …

Question number: 43

» Probability » Definitions and Axiomatic Approach

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let X be a random variable defined on (Ω, A, P). Define a point function F (x) =P {ω: X (ω) ≤ x}, for all xϵR. Shoe that the function F is indeed a distribution function.

Explanation

Let x 1 < x 2. Then (-∞, x 1] Equation ( (-∞, x 2] and we have

Equation

Since F is non decreasing, it is sufficient show that for any sequence of numbers x n ↓x, x 1 > x 2 > … > x n

… (86 more words) …

Question number: 44

» Probability » Expectation

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that E (X - a) 2 is minimized for a = E (X), assuming that the· first 2 moments of X exist.

Explanation

Assume Y = E (X - a) 2

Here we want to minimized Y for a that is

Equation

Equation

Equation

Equation

Equation

Equation

The value of E (X - a) 2 is minimum when the value of a is E (X).

Question number: 45

» 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

f Page
Sign In