Question

Response times when input is sent from a wired terminal are (generally) much faster. In this situation the distribution is Gamma with shape parameter 1.2 and scale parameter 2.5 sec.

a. What’s the probability the response time on the terminal is no more than 8.43 seconds?

b. What’s the probability the response time on the terminal is more than 7.5 seconds?

c. What’s the probability the response time is between 3.5 and 4.5 seconds?

d. Determine the values of the median (50th percentile) of the response time distribution.

e. Determine values l (for “low”) and h (“high”) such that 95% of all response times fall between l and h, with equal probability of a result below l and above h.

f. How long (seconds) is the interval from l to h? The interval from 0 to 8.43 (see part a) also has 0.95 probability; its length is 8.43 sec. It is not necessarily the case that the shortest interval covering a given probability is the one with equal probability in each of the two “tails.” PLEASE USE R AND APPROXIMATE WITHIN 0.0001 PLEASE :^)

Answer 1

As written in the question, I will be using R to solve these questions. I will be providing the R codes and the answers to each of the questions. As directed, the answers will be rounded to four decimal places.

(a) R code: pgamma(8.43,shape=1.2,scale=2.5).
Answer = 0.9500.

(b) R code: 1 - pgamma(7.5,shape=1.2,scale=2.5).
Answer = 0.0713.

(c) R code: pgamma(4.5,shape=1.2,scale=2.5) - pgamma(3.5,shape=1.2,scale=2.5).
Answer = 0.0971.

(d) R code: qgamma(0.5,shape=1.2,scale=2.5).
Answer = 2.2198.

(e) R code: h = qgamma(0.975,shape=1.2,scale=2.5).
Answer = 10.2434.
R code: l = qgamma(0.025,shape=1.2,scale=2.5).
Answer = 0.1283.

(f) R code: qgamma(0.975,shape=1.2,scale=2.5) - qgamma(0.025,shape=1.2,scale=2.5).
Answer = 10.1152.