Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 52 - 56 of 72

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Question number: 52

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2014

Essay Question▾

Describe in Detail

If f x (x) be the probability density function of a N (µ, σ 2) distribution, then show that

LUxf(x)dx=μ[Φ(U)Φ(L)]σ[ϕ(U)ϕ(L)]

where L=Lμσ , U=Uμσ and φ (x) and Φ (x) are the probability density function and distribution function of the standard normal distribution respectively.

Explanation

Given that f X (x) be the probability density function follows N (µ, σ 2).

fX(x)=12πσe12(xμσ)2

Then

LUxf(x)dx=… (216 more words) …

Question number: 53

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2015

Essay Question▾

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Let X follow log-normal with parameters µ and σ 2. Find the distribution of Y = aX b, a > 0, -∞ < b < ∞

Explanation

If X follow log-normal with parameters µ and σ 2, then Z = logX follow normal distribution.

f(x)=12πσxe12(logxμσ)2,0<x<

f(… (341 more words) …

Question number: 54

» Probability » Characteristic Function

Appeared in Year: 2009

Essay Question▾

Describe in Detail

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: 55

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2014

Essay Question▾

Describe in Detail

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: 56

» Probability » Conditional Probability

Appeared in Year: 2009

Essay Question▾

Describe in Detail

(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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