ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 50 - 56 of 164
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Question 50
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 51
Appeared in Year: 2015
Describe in Detail
Essay▾By using Euler-Maclaurin formula, find the sum
Explanation
The Euler-Maclaurin formula is
where
Given the equation
Question 52
Appeared in Year: 2014
Describe in Detail
Essay▾Let X1 , X2 , … , Xn be n independently identically distributed random variables following an exponential distribution with mean parameter θ. Obtain the distributions of (i) Maximum (X1 , X2 , … , Xn) ; (ii) Minimum (X1 , X2 , … , Xn) and (iii) Median for n = 2m + 1, (m > 0)
Explanation
Let X1 , X2 , … , Xn , be n independently identically distributed random variables following an exponential distribution with mean parameter θ. The probability density and distribution function is
Let X(1) , X(2) , … , X(n) denote the order statistic of a random sample, X1 , X2 , … , Xn from exponential distribution with mean parameter θ. Then the d…
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Question 53
Appeared in Year: 2015
Describe in Detail
Essay▾Obtain the characteristic function of X whose pdf is
Explanation
The characteristic function of X is
Let assume x-µ = v, then dx = dv
Assume that , then dv = λ du
First find the value of this integral, this solution adopts the method of contour integration.
Let assume t > 0, For every n we define in the complex plane the contour Cn is ni
By the residue theorem in the theory of complex we have the function
has in the …
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Question 54
Appeared in Year: 2013
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Essay▾Let X be a continuous random variable and have F (x) as the distribution function. If E [X] exists, then show that:
Explanation
We known that in continuous random variable, the mean is
We know that
dF (x) =-dS (x)
Hence it՚s proofed.
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Question 55
Appeared in Year: 2010
Describe in Detail
Essay▾The following data represent lifetimes (hours) of batteries for two different brands:
Brand A | 40 | 30 | 40 | 45 | 55 | 30 |
Brand B: | 50 | 50 | 45 | 55 | 60 | 40 |
Test whether two brands are the same.
Explanation
we use Wilcoxon two-sample test to the following data
Let n1 be the number of breakdown times in the Brand A with median time m1 , and n2 the number of breakdown times in the Brand B with median time m2 . Then the hypothesis is that the median times of the two brands are same against the alternative that they are not same.
For this first arrange the …
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Question 56
Appeared in Year: 2012
Describe in Detail
Essay▾Let X1 , X2 , ···, Xn be random variables such that
Also
Find
Explanation
To find the mean and variance, we use the conditional expectation and conditional variance
By the given definition of conditional expectation
.
.
.
The variance is