# Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 38 - 43 of 72

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## Question number: 38

» Probability » Elements of Measure Theory

Appeared in Year: 2014

### Describe in Detail

Suppose that all the four outcomes 0 _{1}, 0 _{2}, 0 _{3} and 0 _{4} of an experiment are equally likely. Define A = (0 _{1}, 0 _{4}), B = (0 _{2}, 0 _{4}) and C = (0 _{3}, 0 _{4}). What can you say about the pairwise independence and mutually independence of the events A, B and C?

### Explanation

Given that there are four outcomes. Define A = (O _{1}, O _{4}), B = (O _{2}, O _{4}), C = (O _{3}, O _{4})

Their intersection is and Union is AUBUC = (O _{1},… (193 more words) …

## Question number: 39

» Probability » Tchebycheffs Inequality

Appeared in Year: 2015

### 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

or … (76 more words) …

## Question number: 40

» Probability » Elements of Measure Theory

Appeared in Year: 2012

### 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

… (82 more words) …

## Question number: 41

» Probability » Standard Probability Distributions » Binomial

Appeared in Year: 2010

### Describe in Detail

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

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} +… (263 more words) …

## Question number: 42

» Probability » Standard Probability Distributions » Gamma

Appeared in Year: 2011

### Describe in Detail

Let

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

… (380 more words) …

## Question number: 43

» Probability » Definitions and Axiomatic Approach

Appeared in Year: 2010

### 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}] ( (-∞, x _{2}] and we have

Since F… (264 more words) …