MATH 3081
Name
Homework 15
.
5.2.1 A random sample of size 8-X1 = 1, X2 = 0, X3 = 1, X4 = 1, X5 = 0, X6 = 1, X7 = 1, and X8 = 0-is taken from
the probability function
k
1−k
PX (k; θ) = θ (1 − θ) , k = 0, 1; 0 < θ < 1.
Find the maximum likelihood estimate for θ.
5.2.3 Use the sample Y1 = 8.2, Y2 = 9.1, Y3 = 10.6, and Y4 = 4.9 to calculate the MLE for λ in the exponential pdf
fY (y; λ) = λe−λy , y ≥ 0.
5.2.4 Suppose a random sample of size n is drawn from the probability model
2
PX (k; θ) =
θ2k e−θ
, k = 0, 1, 2, . . .
k!
Find a formula for MLE, θ̂.
5.2.5 Given that Y1 = 2.3, Y2 = 1.9, and Y3 = 4.6 is a random sample from
y
y 3 e− θ
fY (y; θ) =
, y ≥ 0.
6θ4
Calculate the MLE for θ.
5.2.6 Use the method of MLE to estimate θ in the pdf
√
θ
fY (y; θ) = √ e− y θ , y > 0.
2 y
Evaluate θ̂. for the following random sample of size 4: Y1 = 6.2, Y2 = 7, Y3 = 2.5, and Y4 = 4.2.
5.2.7 An engineer is creating a project scheduling program and recognizes that the tasks making up the project are not
always completed on time. However, the completion proportion tends to be fairly high. To reflect this condition,
he uses the pdf
fy (y; θ) = θ y θ − 1 , 0 ≤ y ≤ 1, θ > 0
where y is the proportion of the task completed. Suppose that in his previous project, the proportions of tasks
completed were 0.77, 0.82, 0.92, 0.94, and 0.98. Estimate θ.
5.2.10 (a) Based on the random sample Y1 = 6.3, Y2 = 1.8, Y3 = 14.2, and Y4 = 7.6, use the method of maximum
likelihood to estimate the parameter θ in the uniform pdf
fy (y; θ) =
1
,
θ
0 ≤ y ≤ θ.
(b) Suppose the random sample in part (a) represents the two-parameter uniform pdf
fy (y; θ1 , θ2 ) =
1
,
θ2 − θ1
θ1 ≤ y ≤ θ2 .
Find the maximum likelihood estimates for θ1 and θ2 .
5.2.11 Find the maximum likelihood estimate for θ in the pdf
fy (y; θ) =
2y
,
1 − θ2
θ ≤ y ≤ 1,
if a random sample of size 6 yielded the measurements 0.70, 0.63, 0.92, 0.86, 0.43, and 0.21.