# 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

### 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}

## Question number: 94

» Statistical Methods » Correlation Coefficient » Partial Correlation

Appeared in Year: 2009

### 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

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

## Question number: 95

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2013

### Describe in Detail

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

= 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

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

The marginal mass functions of Y is

## Question number: 96

» Numerical Analysis » Interpolation Formulae » Gauss

Appeared in Year: 2011

### Describe in Detail

Use mathematical induction to prove

### 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

## Question number: 97

Appeared in Year: 2011

### 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

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

### Describe in Detail

Let X have pdf

Obtain the cdf of Y = X ^{2}.

### Explanation

## Question number: 99

» Statistical Methods » Standard Errors and Large Sample Tests

Appeared in Year: 2009

### 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