ISS Statistics Paper I (Old Subjective Pattern): Questions 120 - 124 of 165

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Question number: 120

» Statistical Methods » Correlation Coefficient » Multiple Correlation

Appeared in Year: 2015

Essay Question▾

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With 3 variables X 1, X 2 and X 3, it is given that r 13 =0.71, R 1.23 =0.78. Find r 12.3.

Explanation

We know that

r12.3=r12r13r23(1r132)(1r232)

R1.23=r122+r1322r12r13r231r232… (95 more words) …

Question number: 121

» Statistical Methods » Correlation Coefficient » Intraclass Correlation

Appeared in Year: 2012

Essay Question▾

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For a set of 10 pairs of observations (x i, y j), i = 1 (1) 10, the following calculations are available

xi=61,xi2=412,yi=45,yi2=308,xiyi=295

Examine at 5 % level of significance if the two variables arc uncorrelated in the population.

Explanation

First, we calculate the sample correlation coefficient r

r=(xix)(yiy)(xix)2(yiy)2

for i = 1 to 10

r… (173 more words) …

Question number: 122

» Probability » Characteristic Function

Appeared in Year: 2009

Essay Question▾

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Find the density, if its characteristic function is

ϕ(t)={1 t , t 10,otherwise

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) …

Question number: 123

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2014

Essay Question▾

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If f x (x) be the probability density function of a lognormal distribution, show that

LUxkf(x)dx=ekμ+12k2σ2[Φ(Uk)Φ(Lk)]

Where Lk=logLμσkσ and upper limit is Uk=logUμσkσ and φ (z) is the distribution function of the standard normal distribution. Hence find E (X) and V (X).

Explanation

Given that f X (x) has a lognormal distribution, the probability density function is

fX(x)=1x2πσe12(logxμσ)2

Then

LUxkf(x)d… (434 more words) …

Question number: 124

» Probability » Conditional Probability

Appeared in Year: 2009

Essay Question▾

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(i) Let X be a random variable such that P [X < 0] = 0 and E [x] exist. Show that P (X ≤ 2E [x] ) ≥ l/2

(ii) Let E [X] = 0 and E [X 2] be finite. Show that P (X 2 < 9E [X 2] ) > 8/9

Explanation

(i) Using Markov inequality, for any random variable and constant a > 0

P[Xa]E(X)a

Here a = 2E (X)

P[X2E(X)]E(X)2E(X)… (177 more words) …

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