Question

Mrs. Carrie works as a lead teacher at a preschool. She has decided to send an email asking for volunteers to come and read stories to the children. Mrs. Carrie sends the email to 20 parents. Suppose that the time it takes each of the parents to respond is an independent exponential random variable with mean 15 minutes. Mrs. Carrie will stop checking emails after she gets 6 responses, and she constantly monitors her email until she gets those 6 responses. If T is the amount of time Mrs. Carrie is monitoring her emails, (a) (10 points) Find E[T]. (b) (10 points) Find Var(T).

Answer 1

(a) E[T] = 90 minutes

(b) Var(T) = 225 minutes²

Step-by-step explanation:

(a) The expected time for one parent to respond is given as the mean of an exponential random variable, which is 15 minutes. Since Mrs. Carrie is waiting for 6 responses, the expected time for her to get these 6 responses is calculated as the sum of the individual expected times: 15 minutes * 6 = 90 minutes.

(b) To find the variance of T, we use the property of the exponential distribution where the variance of an exponential random variable with mean μ is equal to μ². The variance for one parent's response time is 15² = 225 minutes². Since the response times for each parent are independent, the variance for all 6 parents is the sum of the variances, resulting in a total variance of 225 * 6 = 1350 minutes².

Second method

(a) Each parent's response time is an independent exponential random variable with a mean of 15 minutes. The expected time for Mrs. Carrie to receive one response is the mean, 15 minutes. As she needs 6 responses, the expected total time is calculated by multiplying the expected time for one response by the number of responses needed: 15 minutes * 6 = 90 minutes.

(b) For an exponential distribution, the variance is equal to the square of the mean. The variance for one parent's response time is 15² = 225 minutes². Since the response times for each parent are independent, to find the total variance for 6 parents, we multiply the variance for one parent by the number of parents: 225 minutes² * 6 = 1350 minutes².

Third method

(a) Considering each parent's response time as an independent exponential random variable with a mean of 15 minutes, the expected time for Mrs. Carrie to obtain 6 responses is found by multiplying the mean response time by the number of responses needed, resulting in an expected time of 90 minutes.

(b) The variance of an exponential distribution with mean μ is μ². Therefore, for one parent's response time, the variance is 15² = 225 minutes². Extending this to 6 parents, due to the independence of their response times, the total variance is determined by multiplying the variance for one parent by the number of parents, yielding a total variance of 1350 minutes².