Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 1 - 8 of 72

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Question number: 1

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that the square of the one sample t-statistic has the F-distribution. What are its degrees of freedom?

Explanation

The t-statistic is defined as the ratio of a standard normal variable X~N (0, 1) and the square root of Y/n where Y~ χn2 and n is the degree of freedom.

t=XYn

Then we show the square of t-statistic follows F-distribution.

t… (430 more words) …

Question number: 2

» Probability » Central Limit Theorems

Appeared in Year: 2013

Essay Question▾

Describe in Detail

State and prove Lindeberg-Levy Central limit theorem.

Explanation

Lindeberg-Levy Central limit theorem.

Let Y 1, Y 2, …, Y n be independent and identically distributed random variables with common mean

E (Y i) =µ and finite positive variance Var (Y i) = σ 2 for i = 1, 2, …, n

then n… (172 more words) …

Question number: 3

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2013

Essay Question▾

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If X ~ N (0, 1), obtain the distribution of X 2.

Explanation

X ~ N (0, 1). The density function is

f(x)=12πex22

Let assume Y = X 2

F(y)=P[Yy]=P[X2y]

=P[… (139 more words) …

Question number: 4

» Probability » Expectation

Appeared in Year: 2014

Essay Question▾

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In a lottery 1000 tickets are sold and the cost of a ticket is if 10. The lottery offers a first prize of if 1, 000, two second prizes of if 500 each, and three third prizes of if 100 each. A person purchases a ticket. If X denotes the amount he may get, find E (X) and V (X).

Explanation

The probability of purchases a ticket is

P=11000

First prize amount is 1000

Two second prizes amount is 500 each

Three third prizes amount is 100 each

X denotes the amount he may get

Then, the expected value of X is

E(X)=1000… (90 more words) …

Question number: 5

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Prove that for among the discrete distributions, the geometric distribution has the lack of memory property.

Explanation

The property of memory less is that these distributions of “time from now to the next period” are exactly the same. The property is most easily explained in terms of “waiting times.

Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is… (315 more words) …

Question number: 6

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

Essay Question▾

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Prove that the sum of two independent chi-squared random variables is also chi-squared.

Explanation

Let X and Y are two independent chi-squared random variables with degree of freedom n and m respectively. We proof this by moment generating function. The moment generating function of chi-squared distribution is

MX(t)=(12t)n2

Then moment… (134 more words) …

Question number: 7

» Probability » Probability Generating Functions

Appeared in Year: 2013

Essay Question▾

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Let P (z) be the probability generating function of the random variable X whose probability distribution is Pr [X = n] = p n, n = 0, I, 2, …… Find the generating function of (i) Pr [X > n],

(ii) Pr [X < n] and (iii) Pr [X = 2n].

Explanation

Let P (z) be the probability generating function of the random variable X whose probability distribution is Pr [X = n] = p n. Then

P(z)=n=0znpn

(i) The generating function of Pr [X > n] is… (475 more words) …

Question number: 8

» Probability » Tchebycheffs Inequality

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Show that for 40, 000 throws of a balanced coin, the probability is at least 0.99 that the proportion of heads will fall between 0.475 and 0.525.

Explanation

For balanced coin the probability is p = 0.5. For Bernoulli trails where n = 40, 000, the mean and standard deviation is

μ=40, 000×0.5=20, 000

σ=40, 000×0.5×0.5=100

Using Chebyshev’s inequality,

P(Xμ… (83 more words) …

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