Question

3. This problem is concerned with the maximum likelihood estimate (MLE) of various distributions. Bob, Céline and Daisy want to model the distribution of the heights of 20 students in the classroom. They get the following data (in cm) : 168, 177, 194, 169, 159, 172, 174, 177, 159, 172, 181, 171, 168, 162, 168, 157, 180, 174, 162, 177. (i) Bob took Math170A, and he wants to model the heights by the normal distribution with probability density p(x) e 202 for- 0o < oo where (μ, σ2) are two parameters to estimate. What are the MLE of (μ, σ2)? (ii) Céline did not take any course in science, and she wants to model the heights by the uniform distribution with probability density p(x)for 0 3<0, where θ is a parameter to estimate. What is the MLE of θ? iii) Daisy took an advanced class in statistics, and she wants to model the heighs by the double exponential distribution with probability density where ξ is a parameter to estimate. What is the MLE of ξ? Which of the above models is/are more reliable? Which of the above estimations is/are simple?

Answer 1

168
177
194
169
159
172
174
177
159
172
181
171
168
162
168
157
180
174
162
177
mean	171.05
variance	79.73421
maximum	194
median	171.5

a)

form normal distribution

sample mean and sample variance are MLE for $\mu$ and $\sigma^2$

MLE of   $\mu$ = 171.05

MLE of $\sigma^2$= 79.73421

b),

for Uniform distribution

MLE for $\theta$ is Max(X1,X2,...Xn)

= 194

c)

for double exponential distribution

MLE of $\xi$ is Median = 171.5