You are operating a train. Ticket for this train costs $10. The train is late to

Question

You are operating a train. Ticket for this train costs $10. The train is late to the destination T minutes, where T is an exponential random variable with parameter = 4. If the train is more than 2 minutes and less than 4 minutes late then each customer gets half of the ticket refunded. If the train is more than 4 and less than 8 minutes late each customer gets the full price of the ticket refunded. If the train is more than 8 minutes late each customer gets the full price of the ticket refunded and in addition gets $n.

(a) For what values of n is the expectation of your profit positive?

(b) What is the probability you earn at least $3 per ticket?

Explanation / Answer

Solution-

We calculate the following properties-

P( 2 < T < 4) = P( T < 4) - P(T < 2)

= e-2*4 - e-4*4

= 0.0003354

P( 4 < T < 8) = P( T < 8) - P(T < 4)

= e-4*4 - e-8*4

= 1.12589E-07

P(T > 8) = e-8*4

= 1.26764E-14

(a) Now E(P) = 10 - [5 * P(2 < T < 4 ) + 10 * P( 4 < T < 8) + (10+n) * P( T >8) ]

=10 - 0.001678 - 1.26764E-14 * n

If this is positive, then

n > 788736915037117

Thanks!

n > 788736915037117

Thanks!

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