MLF week 12 - real exam questions

Let be a discrete random variable such that with where is an unknown parameter. Consider the random sample from . The maximum likelihood estimate of is:

The dataset is sampled from a distribution whose PDF is given below with parameter : Assume that each point in the dataset is sampled independently from this distribution. Find the maximum likelihood estimate for . Your answer should be an integer.

Let be a discrete random variable such that and where is an unknown constant. Consider a random sample . Find the maximum likelihood estimate of for the given sample. Enter the answer correct to one decimal place.

The time (in minutes) a bank employee spends assisting a customer follows an exponential distribution with parameter . The time spent (in minutes) from a random sample of 5 customers are as follows: Find the maximum likelihood estimate of for the given sample. Enter the answer correct to two decimal places.

In a large town, it is estimated that 20% of students own a laptop. A random sample of 400 students is selected from the town. What is the variance of the number of students with the laptops in a given sample?

A quality control manager at a factory measures the thickness of glass sheets produced. Five randomly selected sheets have the thicknesses(in mm) 10, 8, 11, 9, 12 respectively. Assuming the thickness follows a normal distribution and the variance is known, what is the maximum likelihood estimator (MLE) of the mean thickness, (μ), in mm of the glass sheets?