# Probability-Definitions and Axiomatic Approach (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 2 of 2

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## Question number: 1

» 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 fo

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## Question number: 2

» Probability » Definitions and Axiomatic Approach

Appeared in Year: 2013

### Describe in Detail

Let X have the density function,

(i) Find the constant c.

(ii) Find the distribution function.

(iii) Compute P [X > -1/2].

### Explanation

(I) To find the value of c using this density function, the integral of density is equal to one by probability definition. The range of X is|X| < 1

when X is positive, | X|=X < 1 and X is negative, | X|=-X < 1 that is X > -1

(ii) The distribution functio

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