1. ## Probability question!

Hey guys,

Urgent help needed!

A computerized switch board in the telephone exchange receives an average 4 calls per minute. Find the probability that –

(a) the interval of time between successive calls at any time is less than 10 seconds.

(b) at least 5 calls are received in one minute.

Thanks a million!

Hey guys,

Urgent help needed!

A computerized switch board in the telephone exchange receives an average 4 calls per minute. Find the probability that –

(a) the interval of time between successive calls at any time is less than 10 seconds.

(b) at least 5 calls are received in one minute.

Thanks a million!
a) It sounds like an exponential distribution with mean $\displaystyle \frac{1}{4}$ (minutes/calls). So, the probability the interval of time between calls is less that 10 seconds (or equivalently, 1/6 of a minute) is

$\displaystyle P(Y<\frac{1}{6})= \int_{0}^{\frac{1}{6}}\frac{1}{4}e^{-\frac{y}{4}}dy$.

b) Now, instead of the time between events we are interested in the number of events in a given time interval. So, your random variable Y is poisson with mean $\displaystyle \lambda=4$. So, the probability is

$\displaystyle P(Y\geq\\5)=\sum_{y=5}^{\infty}$$\displaystyle \frac{\lambda^{y}e^{-\lambda}} {y!}$.
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