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April 24th, 2017, 01:03 AM   #1
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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.
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April 24th, 2017, 01:46 AM   #2
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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
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April 27th, 2017, 03:57 AM   #3
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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)
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April 27th, 2017, 08:34 AM   #4
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Quote:
Originally Posted by shashank dwivedi View Post
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?
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April 27th, 2017, 08:50 AM   #5
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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?
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April 27th, 2017, 08:59 AM   #6
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Quote:
Originally Posted by shashank dwivedi View Post
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.
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