ISS Statistics Paper I (Old Subjective Pattern): Questions 53 - 58 of 165

Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to 165 questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.

Rs. 550.00 or

Question number: 53

» Probability » Characteristic Function

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Obtain the characteristic function of X whose pdf is

f(x)=λπ.1λ2+(xμ)2,<x<

Explanation

The characteristic function of X is

E(eitx)=λπeitxλ2+(xμ)2dx

Let assume x-µ = v, then dx = dv

=λπ… (403 more words) …

Question number: 54

» Probability » Expectation

Appeared in Year: 2013

Essay Question▾

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:

E(X)=0(1F(x))dx0F(x)dx

Explanation

We known that in continuous random variable, the mean is

E(X)=xf(x)dx

=0xf(x)dx+0xf(x)dx

We… (135 more words) …

Question number: 55

» Statistical Methods » Non-Parametric Test » Wilcoxon

Appeared in Year: 2010

Essay Question▾

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… (284 more words) …

Question number: 56

» Probability » Expectation

Appeared in Year: 2012

Essay Question▾

Describe in Detail

Let X 1, X 2, ···, X n be random variables such that

E(X1)=μ,E(Xr Xr1)=Xr1;r=2(1)n

Also

E(X1μ)2=σ2,E[(XrXr1)2 Xr1]=σ2;r=2(1)n

Find E(Xn)andVar(Xn)

Explanation

To find the mean and variance, we use the conditional expectation and conditional variance

E(Xn)=E[E(XnXn1)]

By the given definition of conditional expectation

=E(Xn1)

=E[… (194 more words) …

Question number: 57

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let X 1, X 2, …, X n be independent N (0, σ 2) random variables. Obtain the mean and variance of i=1nXi2 . What is its probability distribution?

Explanation

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

f(x)=12πσex22σ2

First we find the distribution of Y = X 2

F(y)=P[Yy]=… (243 more words) …

Question number: 58

» Probability » Characteristic Function

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Find the density function of X whose characteristic function Φ (t) is given by:

ϕ(t)={1 t , t 10, t >1

Explanation

Rewrite the characteristic function Φ (t)

ϕ(t)={1t,0<t11+t,1<t<00,t>1ort<1

ϕ(t)={1… (236 more words) …

Sign In