It is observed that during a specific time period of any wee

It is observed that during a specific time period of any weekday, the number of clicks at a popular website in any 1minute interval is a Poisson random variable, N, with an average of 100 clicks.  

(a) Suppose that if there is more than 1000 clicks in a 1minute interval, users will experience noticeable slow response. What is the probability that the response time from the web server is satisfactory (i.e., no noticeable slow response time)? (Note: For part a), you don’t need to give the exact numerical result, a formula will be sufficient.)

(b) What is the probability that during this time period there is no more than one click at this website in 1 second?

Solution

a)

Note that the probability of x successes out of n trials is  
  
P(x) = u^x e^(-u) / x!  
  
where  
  
u = the mean number of successes =    100
  
x = the number of successes

Hence,

P(x>1000) = 1 - Sum(P(x))|(x=0 to x = 1000) [ANSWER]

where P(x) is defined above.

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b)

There are 100 clicks in 60 s, so an average of 100/60 = 1.666667 clicks per second.

Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    1.666666667      
          
x = the maximum number of successes =    1      
          
Then the cumulative probability is          
          
P(at most   1   ) =    0.503668274 [ANSWER]