Probability-Standard Probability Distributions [ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)]: Questions 1 - 7 of 22
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Question 1
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.
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Question 2
Appeared in Year: 2013
Describe in Detail
Essay▾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 said to have lack of memory if
Geometric distribution …
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Question 3
Appeared in Year: 2011
Describe in Detail
Essay▾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 X and Y are independent
which is the moment generating fu…
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Question 4
Appeared in Year: 2012
Describe in Detail
Essay▾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 r and proportion of item which is defect that is probabil…
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Question 5
Appeared in Year: 2010
Describe in Detail
Essay▾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,
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Question 6
Appeared in Year: 2011
Describe in Detail
Essay▾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 = , then x =-1
For median q = , then x = 0
For third quartile q = , then x = 1
Question 7
Appeared in Year: 2011
Describe in Detail
Essay▾Let X1 , X2 , … , Xn 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 gamma distribution with parameter .
The additive property of gamma distribution
Then, the mean and variance of gamma distribution is
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