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

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## 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

σ ^{2} = E [X ^{2}] - (E [X] ) ^{2} =4

Then, a lower bound for the probability

Using Chebyshev’s inequality

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## 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

The events are independent because only one event happens

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## 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} + X _{2} + …. + X _{m}, sum of random variables are found by moment generating function. The moment generating function of binomial distribution is

So, the …

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## 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

Assume but limit is same

This integral is a gamma function

So,

Thus f (x) is a probability density function.

To find the variance, we first find the k ^{th} moment

Since the integrate is th…

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## 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 is non decreasing, it is sufficient show that for any sequence of numbers x _{n} ↓x, x _{1} > x _{2} > … > x _{n} > …x, F (x _{n}) →F (x). Let A _{k} = {ω, X (ω) ϵ (x, x _{k}] }. Then A _{k} ϵS and also

Since none of the interval (x, x _{k}] contains x. It follows that but

So that

and F is…

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## Question number: 44

» Probability » Expectation

Appeared in Year: 2011

### 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

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

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## Question number: 45

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

### Describe in Detail

Let X have pdf

Obtain the cdf of Y = X ^{2}.

### Explanation

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