Question

1. (a) Y1,Y2,...,Yn form a random sample from a probability distribution with cumulative distribution function FY (y) and probability density function fY (y). Let Y(1) = min{Y1,Y2,...,Yn}. Write the cumulative distribution function for Y(1) in terms of FY (y) and hence show that the probability density function for Y(1) is fY(1)(y) = n{1−FY (y)}n−1fY (y). [8 marks]

(b) An engineering system consists of 5 components connected in series, so, if one components fails, the system fails. The lifetimes (measured in years) of the 5 components, Y1,Y2,...,Y5, are all independent and identically distributed. (i) Suppose the lifetimes follow the standard uniform distribution U(0,1). Find the probability density function for Y(1), the time to failure for the system, and hence ﬁnd the probability that the system functions for at least 6 months without failing. [10 marks] (ii) If, instead, the lifetimes follow an exponential distribution with mean θ, then Y(1) follows an exponential distribution with mean θ/5. Prove this result. Assuming that the only information available is a single observation on Y(1), ﬁnd the most powerful test of size 0.05 for H0 : θ = θ1 versus H1 : θ = θ2, where θ1 < θ2. (Hint: the probability density function and cumulative distribution function for an exponential random variable with mean θ are f(y) = θ−1 exp(−y/θ), y > 0, and F(y) = 1−exp(−y/θ), y > 0, respectively.) [12 marks]

Answer #1

Let Y1,Y2,...,Yn denote a
random sample of size n from a population with a uniform
distribution on the interval (0,θ). Let Y(n)=
max(Y1,Y2,...,Yn) and U =
(1/θ)Y(n) .
a) Show that U has cumulative density function
0 ,u<0,
Fu (u) = un ,0≤u≤1,
1 ,u>1

Let Y1, Y2, . . ., Yn be a
random sample from a Laplace distribution with density function
f(y|θ) = (1/2θ)e-|y|/θ for -∞ < y < ∞
where θ > 0. The first two moments of the distribution are
E(Y) = 0 and E(Y2) = 2θ2.
a) Find the likelihood function of the sample.
b) What is a sufficient statistic for θ?
c) Find the maximum likelihood estimator of θ.
d) Find the maximum likelihood estimator of the standard
deviation...

Let Y1, Y2, ... Yn be a random sample of an exponential
population with parameter θ. Find the density function of the
minimum of the sample Y(1) = min(Y1, Y2, ..., Yn).

Let Y1,Y2,Y3,...,Yn denote a random sample from a Poisson
probability distribution.
(a) Show that sample mean, ˆ θ1 = ¯ Y and the sample variance, ˆ
θ2 = S2 are both unbiased estimators of θ.
(b) Calculate relative eﬃciency of the two estimators in (a).
Based on your calculation, Which of the two estimators in 3a would
you select as a better estimator?

. Let Y1, ..., Yn denote a random sample
from the exponential density function given by f(y|θ) =
(1/θ)e-y/θ when, y > 0 Find an MVUE of V
(Yi)

Let Y1, Y2, . . ., Yn be a
random sample from a uniform distribution on the interval (θ - λ, θ
+ λ) where -∞ < θ < ∞ and λ > 0. Find the method of
moments estimators of θ and λ.

Suppose we have a random sample Y1, . . . , Yn from a CRV with
density
fY (y; θ) = θ*(y + 1)^(θ+1) where y > 0, θ > 1
Find the MME and MLE for θ.

Let Y1, Y2, . . . , Yn denote a random sample from a uniform
distribution on the interval (0, θ). (a) (5 points)Find the MOM for
θ. (b) (5 points)Find the MLE for θ.

(a) Let Y1,Y2,··· ,Yn be i.i.d.
with geometric distribution P(Y = y) = p(1−p)y-1 y=1, 2,
........, 0<p<1. Find a suﬃcient statistic for p.
(b) Let Y1,··· ,yn be a random sample of size n from
a beta distribution with parameters α = θ and β = 2. Find the
suﬃcient statistic for θ.

Suppose Y1,··· ,Yn is a sample from a
exponential distribution with mean θ, and let Y(1),···
,Y(n) denote the order statistics of the sample.
(a) Find the constant c so that cY(1) is an unbiased
estimator of θ.
(b) Find the suﬃcient statistic for θ and MVUE for θ.

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