Probability 

A motorcyclist passes 2 independent traffic lights, each red, with a probability of 0.25, with waiting times in units of minutes, which are independent random variables, exponentially distributed with parameter lambda=2. We will mark N as the number of traffic the rider stopped,  and X as the total waiting time at the traffic lights.
A. How X|N=n is distributed for n=0,1,2?
B. What is the expectancy of X?
C. What is the probability that less than 2 minutes will be wasted on a random morning?
D. Calculate E[e^(tX)|N=n] for n=0,1,2
E. Find the Moment-generating function of X. Does X is normally distributed?

Probability
Danielatia123 Danielatia123
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  • Mathe Mathe
    +1

    I would suggest a bounty of $7 per question. As it is, the bounty is quite low.

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Martin Martin
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  • Martin Martin
    0

    For the distribution of the sum of two ind. exp. variables see below

    • Martin Martin
      0

      https://en.wikipedia.org/wiki/Exponential_distribution#Sum_of_two_independent_exponential_random_variables

  • Danielatia123 Danielatia123
    0

    I don't want to be petty , but X|N is continues , so actually the probability that X is equal to something is zero , because its a continues variable. No? I mean you should write X<=x|N , no?

    • Martin Martin
      0

      I didn't receive a notification for your comment, so I am not sure when you wrote this. But those are supposed to be density functions, not probabilities, and you are right the probabilities are zero themselves.

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