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

Essay Question▾

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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 ABC=O4 and Union is AUBUC = (O 1,… (193 more words) …

Question number: 39

» Probability » Tchebycheffs Inequality

Appeared in Year: 2015

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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

P(Xμkσ)11k2

or P(Xμ<k)>1… (76 more words) …

Question number: 40

» Probability » Elements of Measure Theory

Appeared in Year: 2012

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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

=P(ABC)

=1P(ABC)c

=1P(… (82 more words) …

Question number: 41

» Probability » Standard Probability Distributions » Binomial

Appeared in Year: 2010

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

P(X=k)=(nk)pk(1p)nk,k=0, 1,2,,n;0<p<1

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

Essay Question▾

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Let

f(x)=1Γnβnxn1exβ,x>0,n>0,β>0

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

0f(x)dx=1

0f(x)dx=01… (380 more words) …

Question number: 43

» Probability » Definitions and Axiomatic Approach

Appeared in Year: 2010

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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

F(x1)=P(Xx1)P(Xx2)=F(x2)

Since F… (264 more words) …

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