Let U1, ....U5 be independent and standard uniform distibuted random variables given by P(U1 ≤ x) = x, 0 < x < 1 1. Compute the moment generating function E(e sU ) of the random variable U1. 2. Compute the moment generating function of the random variable Y = aU1 + U2 + U3 + U4 + U5 with a > 0 unknown. 3. Compute E(Y ) and V ar(Y ). 4. As an estimator for the unknow value θ = a we migth use as an estimator θb = 2 n Xn i=1 Yi − 4 = 2Y − 4. with Yi independent and identically distributed having the same cdf as the random variable Y discussed in part 2. Compute E(θb) and V ar(θb) and explain why this estimator is sometimes not very useful. 5.Give an upperbound on the probability P(| θb− a |> ) for every > 0.(Hint:Use Chebyshevs inequality!) Let U1,..U5 be independent and standard uniform distibuted random variables given byP(U1 < x) = ,0 < x < 1Compute the moment generating function E(esU) of the random variable Uj.Compute the moment generating function of the random variable Y = aU1 +U2 + U3 + U4 + Ug with a > 0 unknown.Compute E(Y) and Var(Y).As an estimator for the unknow value 0 = a we migth use as an estimator>Y; – 4 = 2Y – 4.ni=1with Y; independent and identically distributed having the same cdf as the random variable Ydiscussed in part 2. Compute E(0) and Var(0) and explain why this estimator is sometimesnot very useful.Give an upperbound on the probabilityP(| ô –- a |> e)for every e > 0.(Hint:Use Chebyshevs inequality!)Solution.

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Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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Let U1, ....U5 be independent and standard uniform distibuted random variables given by P(U1 ≤ x) = x, 0 < x < 1 1. Compute the moment generating function E(e sU ) of the random variable U1. 2. Compute the moment generating function of the random variable Y = aU1 + U2 + U3 + U4 + U5 with a > 0 unknown. 3. Compute E(Y ) and V ar(Y ). 4. As an estimator for the unknow value θ = a we migth use as an estimator θb = 2 n Xn i=1 Yi − 4 = 2Y − 4. with Yi independent and identically distributed having the same cdf as the random variable Y discussed in part 2. Compute E(θb) and V ar(θb) and explain why this estimator is sometimes not very useful. 5.Give an upperbound on the probability P(| θb− a |> ) for every > 0.(Hint:Use Chebyshevs inequality!)

Let U1,..U5 be independent and standard uniform distibuted random variables given by
P(U1 < x) = ,0 < x < 1
Compute the moment generating function E(esU) of the random variable Uj.
Compute the moment generating function of the random variable Y = aU1 +
U2 + U3 + U4 + Ug with a > 0 unknown.
Compute E(Y) and Var(Y).
As an estimator for the unknow value 0 = a we migth use as an estimator
>Y; – 4 = 2Y – 4.
n
i=1
with Y; independent and identically distributed having the same cdf as the random variable Y
discussed in part 2. Compute E(0) and Var(0) and explain why this estimator is sometimes
not very useful.
Give an upperbound on the probability
P(| ô –
- a |> e)
for every e > 0.(Hint:Use Chebyshevs inequality!)
Solution.

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