My Math Forum Exponential Distribution problem

 April 24th, 2017, 01:03 AM #1 Member   Joined: Apr 2017 From: India Posts: 75 Thanks: 0 Exponential Distribution problem Calls arrive at a switchboard following an exponential distribution with parameter $\lambda$=5 per hour. If we are at the switchboard, what is the probability that the waiting time for a call is 1) at least 15 minutes (Answer: 0.2865) 2)not more than 10 minutes (Answer: 0.6565) 3)exactly 5 minutes (Answer:0){I think this is because the function is a continuous one) My attempt: Firstly formed the Exponential Distribution Function: F(x)= 1 - $e^{-5x}$ Then for finding the probability : P[X$\ge$15] I am confused at this step. How to approach further? Similarly for second question I am unable to proceed further.For third question , I have given my explanation in the curly brackets above. Please help me out.
 April 24th, 2017, 01:46 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,591 Thanks: 1435 if the call rate is exponential the distribution of the number of calls in a period of time is Poisson with the same rate parameter. 1) You want the probability of 0 calls in $\dfrac 1 4~hr$ $P[0] = \dfrac{\left(\frac 5 4\right)^0 e^{-\frac 5 4}}{0!} = e^{-\frac 5 4} = 0.2865$ you should be able to do (2) and (3) now
 April 27th, 2017, 03:57 AM #3 Member   Joined: Apr 2017 From: India Posts: 75 Thanks: 0 As per me , answer of second question must be: P [X<10] = 1- P [X>=10] =1 - e^(-5/6) =0.5654 ( But it is not matching as answer is 0.6565)
April 27th, 2017, 08:34 AM   #4
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 Originally Posted by shashank dwivedi As per me , answer of second question must be: P [X<10] = 1- P [X>=10] =1 - e^(-5/6) =0.5654 ( But it is not matching as answer is 0.6565)
I get the same answer. Are you sure the given answer is correct?

 April 27th, 2017, 08:50 AM #5 Member   Joined: Apr 2017 From: India Posts: 75 Thanks: 0 Well, the source I am looking is trustworthy, still there can be mistakes. But why the third answer is zero{(3)exactly 5 minutes}, is it because the distribution function is continuous or is there any other explanation?
April 27th, 2017, 08:59 AM   #6
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 Originally Posted by shashank dwivedi Well, the source I am looking is trustworthy, still there can be mistakes. But why the third answer is zero{(3)exactly 5 minutes}, is it because the distribution function is continuous or is there any other explanation?
you are correct about this. the probability of a single point in a continuous distribution is 0.

### Calls arrive at switchboard following an exponential distribution with 5 per hour . What is the probability that the waiting time for a call is at least 15 min , not more than 10 min, exactly 5 min.

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