Probability-Standard Probability Distributions (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 7 of 22

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

Let assume Y = X ²

Let

The limit is also change

Differentiate with respect to y,

‎

So, Y = X ² is follows a chi-square distribution wit

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 said to have lack of memory if

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

Then moment generating function of sum of two random variable (X + Y) is

because X and Y are ind

In this question, the sample size is n = x+r given and each trail only two possible outcomes. The probability of defect is same for each trail and trails are independent. The experiment continues until r defectives.

In the given question the number of manufacturing defect are fixed which is r and proportion of item which is defect that is proba

This proof is a lack of memory property. We known that

Therefore ………… (1)

The equation is

Using equation (1)

Since the equation is true. Therefore,

For find the median and quartile of the Cauchy distribution, q is any quartile. Then, for which value of q, the x value is

For first quartile q = 1/4, then x=-1

For median q = 1/2, then x = 0

For third quartile q = 3/4, then x = 1

Let X ~ N (0, σ ²). The density function is

First we find the distribution of Y = X ²

Let

The limit is also change

Differentiate with respect to y,

‎

So