Question

2. The number of hits on a popular Web page follows a Poisson process with an arrival rate of 5 per minute. One begins observation at exactly noon tomorrow. Let X(1) be the number of hits on the webpage by time t. What is the distribution of the number of hits in the first minute? Give the name of the distribution and the value(s) of parameter(s). What is the probability of 4 or fewer hits in the first minute? What is the distribution of the time until the first hit? Give the name of the distribution and the value(s) of parameter(s). Compute the probability that the time till the first hit exceeds 10 seconds. What is the appropriate distribution for the time until the 4th hit? State the distribution and parameter values. Compute the probability that the time until the 4th hit exceeds 24 seconds. a. b. c.

Answer 1

ANSWER:

Question 2 (a) Here

X(t) be the number of hits on the webpage by time t

so here the distribution of the number of hits in the first minute is poisson distribution with parameter $\lambda$ = 5 * 1 = 5

so here p(x) = e^-5 5^x/x!  ; x >0

Pr(x < 4) = POISSON (x =< 4 ; 5)

by using poissoncdf function we can get this probability.

Pr(x < 4) = POISSONCDF (x =< 4 ; 5) = 0.4405

(b) Here the distribution of the time untile the first hit is an exponential distribution with the parameter $\lambda$ = 1/5 = 0.2 min^-1

so here the pdf is

f(x) = 0.2 e^-0.2x ; x > 0

cdf of the distribution is

F(x) = 1 - e^-0.2x ; x > 0

so here we have to find the probability that the time till the first hit exceeds 10 seconds

Pr(x > 10 second) = 1 - P(x < 10 seconds) = 1- (1 - e^{-0.2* 1/6}) = 0.9672

(c) Here the appropriate distribution for the time untile he 4th hit has gamma distribution with parameter is $\alpha$ = 4 and $\beta$ = 1/ 5 = 0.2

so here as x ~ Gamma( 4, 0.2)

Now we have to find the probability with the help of GAMMADIST function

P(x > 24 seconds) = 1 - GAMMADIST(x < 0.4 mins; 4 ; 0.2) = 1- 0.1429 = 0.8571