Example 7. Let Y1, ... ,Yn be a random sample from a Rayleigh distribution with pdf...
7. Let Y1, ...,Yn be a random sample from the population with pdf f(316) = he=1/0, y>0 (a) Find the MOM estimator for 0. (b) Find the MLE of 0. (c) Find the MLE of P(Y < 2). (d) Find the MLE of the median of the distribution.
3. The Rayleigh distribution is a continuous distribution with pdf of the form Så exp(-+) $(30) = >0 otherwise Suppose that X1,..., X, form a random sample from a Rayleigh distribution where the value of the parameter 8 >0 is unknown. a. Find the maximum likelihood estimator (MLE) of e, assuming that all observed values satisfy 2: >0. b. Is your MLE of 8 a sufficient statistic? Why or why not?
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
Suppose that the population has the following pdf: Le-(y-0) if y> 0 f(y) = { 0 otherwise Let U1 = min{Y1, ... ,Yn} and U2 = $1=1 Yį. (a) Show that the pdf of Uı is f(y) = ne-n(y=0)1(y > 0) (b) Show that U1 - 1/n and U2/n - 1 are both unbiased estimators of 0. (c) Find the variance of each of the unbaised estimators in part (b). (d) One of U1,U2 is a sufficient statistic. Which one?...
If X is Rayleigh distributed amplitude with an average power of 10, what is the pdf of Y-1-exp(-XA2/10)? If X is N(0,4) and Y is N(0,9) and X and Y are independent, what is the prob (XY>0)?
Consider a random sample of size n from a distribution with pdf (In O* S(x; 6) = Ox! x = 0, 1, ...;0 > 1 10 otherwise (a) Find a complete sufficient statistic for 8. (b) Find the MLE of O. (c) Find the CRLB for 6 (d) Find the UMVUE of In e. (e) Find the UMVUE of (In )? (1) Find the CRLB for (In 02
2. Let Yı, ..., Yn be a random sample from an Exponential distribution with density function e-, y > 0. Let Y(1) minimum(Yi, , Yn). (a) Find the CDE of Y) b) Find the PDF of Y (c) Is θ-Yu) is an unbiased estimator of θ? Show your work. (d) what modification can be made to θ so it's unbiased? Explain.
Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf fy(y|0) = (@+ 1) yº, o<y<1 for 0> - -1. (a) Find the MOM estimator of 0. (b) Find the maximum likelihood estimator (MLE) of 0. (c) Find the MLE of the population mean E(Y) = 0 +1 0 + 2 You do not need to prove that the above is true. Just find its MLE.
2. A distribution often used to model the lifetime of electronic components is the Rayleigh density 0 otherwise where θ > 0. (a) If Y has the Rayleigh density, find the distribution for U-Y2 (b) Use the result of part a) to find EY
5. Let X have exponential pdf λe_AE 0 when x > 0 otherwise with λ = 3. Let Y-LX). Find E(Y) and Var(Y)