ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 93 - 99 of 165

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Question number: 93

» Probability » Definitions and Axiomatic Approach

Appeared in Year: 2010

Essay Question▾

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] Equation ( (-∞, x 2] and we have

Equation

Since F is non decreasing, it is sufficient show that for any sequence of numbers x n ↓x, x 1 > x 2 > … > x n

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Question number: 94

» Statistical Methods » Correlation Coefficient » Partial Correlation

Appeared in Year: 2009

Essay Question▾

Describe in Detail

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

Equation

Equation

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

Equation

Equation

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

Equation

Equation

The correlation coefficient is defined by

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Question number: 95

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Let (X, Y) have a joint probability mass function

Equation

= 0, elsewhere

Find the marginal mass functions of X and Y.

Explanation

Let (X, Y) have a joint probability mass function, the individual distribution of either X or Y is called the marginal distribution. So, the marginal mass functions of X is

Equation

and other values of f (x, y) is 0

Equation

Equation

The marginal mass functions of Y is

Equation

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Question number: 96

» Numerical Analysis » Interpolation Formulae » Gauss

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Use mathematical induction to prove

Equation

Explanation

Let ∆ i define the i th finite difference which is defined as

∆ =E-1 where E is a shift operator that is

E n f x = f x + nh

Then

Equation

Equation

Equation

Equation

Question number: 97

» Probability » Expectation

Appeared in Year: 2011

Essay Question▾

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

Equation

Equation

Equation

Equation

Equation

Equation

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

Question number: 98

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Let X have pdf

Equation

Obtain the cdf of Y = X 2.

Explanation

Equation

Equation

Equation

Equation

Equation

Question number: 99

» Statistical Methods » Standard Errors and Large Sample Tests

Appeared in Year: 2009

Essay Question▾

Describe in Detail

A sample of 100 records on lengths of stay of patients in a hospital gave a standard deviation of days of stay as 4.9. In order to estimate the mean number of days of stay within 0.25 day with 95 % confidence, what should be the sample size?

Explanation

Let the patients in a hospital gave a standard deviation of days of stay as 4.9. The mean number of days of stay within 0.25 day that is margin error is the estimate fraction days between the difference of estimated and the true number of days. So, the test statistic

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