Question

1             Let X be an accident count that follows the Poisson distribution with parameter of 3. Determine:                                                                                                                                                                                                            

               a             E(X)                                                                                                                                              

               b             Var(X)                                                                                                                                          

               c             the probability of zero accidents in the next 5 time periods

               d             the probability that the time until the next accident exceeds n

               e             the density function for T, where T is the time until the next accident

                                                                                                                                                     

2             You draw 5 times from U(0,1). Determine the density function for the following random variables:

                                                                                                                                                                                                  

               a             the lowest value                                                                                         

               b             the middle value                                                                                         

               c             the highest value                                                                                                                                                                   

3             You toss a fair coin 1,000 times.                                                                                                          

                                                                                                                                                                                                  

               a             What is the expected # of tails?                                                                                            

               b             What is the variance of the # of tails?                                                                                 

               c             Using the Central Limit Theorem, what is the probability that you will get between 500 and 510 tails?                                                                                                             

                                                                                                                                                     

4             Consider the density function f_X(x) = e^x / (e - 1), with 0 < x < 1. Determine:                                                                                                                                                                                                                               

               a             F(.4)                                                                                                                                             

               b             E(X)       

               c             P(.4 < X < .6)                                                                                                                                                                            

                                                                                                                                                                                   

5             Passengers arrive at a Metro station following a Poisson distribution with parameter 5t (t represents units of time). The train arrival time is distributed on the interval (0,4) according to this density function:

               f(x) = .125 x

Determine:         

               a             E(train arrival time)

               b             E(# of people who will board the train)    

               c             Var(# of people who will board the train)

6             You draw from U(0,1) and define X to be the nearest integer (X will be either 0 or 1). Then you toss a coin 1+X times. Define Y to be the number of tails you get from these tosses. Determine

               a             E(Y)

               b             E(XY)

Answer 1

Question 1:

Here, we are given the distribution as:

a) The mean of the poisson distribution is equal to its parameter. Therefore, we get here:

b) The variance of poisson distribution is equal to its parameter as well. Therefore, we get here:

c) Now the probability of zero accidents in the next 5 time periods is computed here as:

d) The probability that the time until the next accident exceeds n is computed here as:

= Probability of no arrival till time n

e) The waiting time for the next accident is modelled as an exponential distribution with the same parameter as that of the poisson distribution given as: