Probability-Standard Probability Distributions (ISS Statistics Paper I (Old Subjective Pattern)): Questions 1 - 7 of 22
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Question number: 1
» Probability » Standard Probability Distributions » Normal
Appeared in Year: 2013
Describe in Detail
If X ~ N (0, 1), obtain the distribution of X 2.
Explanation
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 with one degree of freedom.
Question number: 2
» Probability » Standard Probability Distributions » Geometric
Appeared in Year: 2013
Describe in Detail
Prove that for among the discrete distributions, the geometric distribution has the lack of memory property.
Explanation
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… (99 more words) …
Question number: 3
» Probability » Standard Probability Distributions » Normal
Appeared in Year: 2011
Describe in Detail
Prove that the sum of two independent chi-squared random variables is also chi-squared.
Explanation
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… (64 more words) …
Question number: 4
» Probability » Standard Probability Distributions » Negative Binomial
Appeared in Year: 2012
Describe in Detail
Items from a large lot are examined one by one until r items with a rare manufacturing defect are found. The proportion of items with this type of defect in the lot is known to be p. Let X denote the number of items needed to be examined. Derive the probability distribution of X, and find E (X).
Explanation
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… (70 more words) …
Question number: 5
» Probability » Standard Probability Distributions » Geometric
Appeared in Year: 2010
Describe in Detail
Let X have a geometric distribution, then for an two non-negative integers m and n,
Prove it
Explanation
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,
Question number: 6
» Probability » Standard Probability Distributions » Cauchy
Appeared in Year: 2011
Describe in Detail
Obtain the median and the quartiles of the Cauchy distribution with p. d. f.
Explanation
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… (7 more words) …
Question number: 7
» Probability » Standard Probability Distributions » Normal
Appeared in Year: 2011
Describe in Detail
Let X 1, X 2, …, X n be independent N (0, σ 2) random variables. Obtain the mean and variance of . What is its probability distribution?
Explanation
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, Y = X 2 is follows a… (24 more words) …