Probability-Standard Probability Distributions (ISS Statistics Paper I (Old Subjective Pattern)): Questions 13 - 16 of 22

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Question number: 13

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2012

Essay Question▾

Describe in Detail

Write down the probability mass function of geometric distribution. State and prove its ‘lack of memory property’. Find also the mean and the variance of the distribution.

Explanation

If there are number of trails such that the probability of success is p. So, the probability that there are x failures before the first success is

f(x)=pqx;x=0, 1,2,

This is the probability mass function… (555 more words) …

Question number: 14

» 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: 15

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2015

Essay Question▾

Describe in Detail

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: 16

» 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) …

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