Question No. 4
I. Suppose we have a random sample $Y_1, Y_2, ..., Y_n$ where:
$Y_j = 0$, if a randomly selected student did not earn a Coursera certificate, and
$Y_j = 1$, if a randomly selected student earned a Coursera certificate.
(5 + 5)
Assuming that the samples ($Y_j$) are independent Bernoulli random variables with an unknown
parameter $p$, find the maximum likelihood estimator of $p$, the proportion of students who earned
a Coursera certificate.
II. Consider the shifted exponential distribution
$f(x) = \lambda e^{\lambda (x - \theta)}$,
$x \ge 0$
When $\theta = 0$, this density reduces to the usual exponential distribution. When $\theta > 0$, there is
positive probability only to the right of $\theta$.
a. Find the maximum likelihood estimator of $\lambda$ and $\theta$ based on a random sample of size $n$.
b. Describe a practical situation in which one would suspect that the shifted exponential
distribution is a plausible model
