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 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, . . ., 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.

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.

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.

(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.

Let Y1, Y2, Y3 ,..,, Yn be a random sample from a normal
distribution with mean µ and standard deviation 1. Then find the
MVUE( Minimum - Variance Unbiased Estimation) for the parameters:
µ^2 and µ(µ+1)

The
joint probability density function for two continuous random
variables is:
f(y1,y2) = k(y1^2 + y2)
for 0 <= y2 <= 1-y1^2
Find the value of the constant k so that this makes f(y1,y2) a
valid joint probability density function.
Also compute (integration) P(Y2 >= Y1 + 1)

f(y1,y2)=k(1−y2), 0≤y1≤y2≤1.joint probability density
function:
a) Find P(Y1≤0.35|Y2=0.3)
b) Find E[Y1|Y2=y2].
c) Find E[Y1|Y2=0.3]

If X is an exponential random variable with parameter λ,
calculate the cumulative distribution function and the probability
density function of exp(X).

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