ISS Statistics Paper I (Old Subjective Pattern): Questions 53 - 58 of 165
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Question number: 53
» Probability » Characteristic Function
Appeared in Year: 2015
Describe in Detail
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… (91 more words) …
Question number: 54
Appeared in Year: 2013
Describe in Detail
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 F (x) =1-S (x) and f (x) =dF (x) /dx
dF (x) =-dS (x)
Hence it’s proofed.
Question number: 55
» Statistical Methods » Non-Parametric Test » Wilcoxon
Appeared in Year: 2010
Describe in Detail
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 n 1 be the number of breakdown times in the Brand A with median time m 1, and n 2 the number of breakdown times in the Brand B with median time m 2. Then the hypothesis is… (191 more words) …
Question number: 56
Appeared in Year: 2012
Describe in Detail
Let X 1, X 2, ···, X n 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
Question number: 57
» 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) …
Question number: 58
» Probability » Characteristic Function
Appeared in Year: 2013
Describe in Detail
Find the density function of X whose characteristic function Φ (t) is given by:
Explanation
Rewrite the characteristic function Φ (t)
The density function of X is given by Fourier inversion theorem using the characteristic function,
The general form is
Putting the characteristic function