ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 88 - 94 of 165

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Question number: 88

» Probability » Elements of Measure Theory

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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

The events are independent because only one event happens

… (110 more words) …

Question number: 89

» Probability » Standard Probability Distributions » Binomial

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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.

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

… (95 more words) …

Question number: 90

» Statistical Methods » Tests of Significance » Chi-Square

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Appeared in Year: 2011

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Explain how to carry out the chi-squared test for on the basis of a random sample

X 1, X 2, …, X n from N (µ, σ 2) population.

Explanation

Given that a random sample X 1, X 2, …, X n from N (µ, σ 2) population. The hypothesis is

The null hypothesis is test by chi-square test only assume the sample size is less than 30.

For this we use the likelihood ratio test

Under the model, the parameter space is

Under the null hypothesis, the parameter spa

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Question number: 91

» Probability » Standard Probability Distributions » Gamma

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Appeared in Year: 2011

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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)

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Question number: 92

» Numerical Analysis » Summation Formula » Euler-Maclaurin's

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Appeared in Year: 2014

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Using Euler’s method, compute the values of y correct upto 4 places of decimal for the differential equation with initial condition x 0 = 0, y 0 = 1, taking h = 0.05.

Explanation

The given differential equation is

with initial condition x 0 = 0, y 0 = 1, taking h = 0.05.

Using Euler’s method,

where

So, putting the initial condition, when n = 0

and

Now first modification of y 1

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Question number: 93

» Probability » Definitions and Axiomatic Approach

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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

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 tha

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Question number: 94

» Statistical Methods » Correlation Coefficient » Partial Correlation

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Appeared in Year: 2009

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The two regression lines between X and Y are 8X - 10Y + 66 = 0,40X - 18Y = 214. The variance of X is 9. Find X, Y, σ Y and ρ.

Explanation

Given that

We know that the mean value of the given series satisfies the regression line that is

The means of X and Y is calculate by multiply equation (1) by 5 to subtract equation (2), we get

The correlation coefficient is defined by

The first equation is written in terms of Y i

… (91 more words) …

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