Question

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

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 fX(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)

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:

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