Question

The exponential distribution has P.D.F. f(t, x):= le-At where >0, and can be a great model for how long one will wait "before the next event happens." Suppose that you're a bank clerk, and on average you wait (1/4) of a minute before someone comes to your window. Suppose, you've already waited 1 whole minute; what's the probability that you'll see someone in the next half a minute?

Answer 1

Exponential is also called the waiting time distribution. This is sometimes tricky to understand so I hope the below explanation helps you in dealing with the similar type of calculations:-

(All calculations below are done in seconds)

The following information is provided:

The provided mean is

B = 15

We need to compute the probability that after having waited already for 60 seconds, we will see someone in the next 30 seconds i.e. between 60 seconds and 90 seconds. In other words, we need to calculate:

Pr(60 < X <90)

Therefore, the following is obtained:

Pr(60 < X <90) = Pr(X <90) - Pr(x <60) = e-60/15--90/15

= 0.0183 -0.0025 = 0.0158

which completes the calculation.

Graphically we have:-

Exponential Distribution - Probability calculations Pr(60 <= x <= 90) = 0.0158 0.080 0.070 - 0.060 - 0.050 - 0.040 - 0.030 - 0.020 - 0.010 - 0.000- 0 10 20 30 40 50 60 70 80 90

Hopefully, this will clear things up. Let me know in the comments if you have any doubts. I will reply ASAP!