ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 89 - 95 of 165

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

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 p

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,

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

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 fo

Given that

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

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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 writ

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

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The marginal mass functions of Y is

and ot