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:

Let X be a random variable which follows a Poisson distribution
whose parameter is equal to 1 . Determine E (X | 4 <X
<14).

Let N have a Poisson distribution with parameter lander=1.
Conditioned on N=n,
let X have a uniform distribution over the integers
0,1,.......,n+1. What is the
marginal distribution for X?
Step by step and show what definition you use

At a train station, international trains arrive at a rate λ = 1
(poisson distribution). At the same train station national trains
arrive at rate λ = 2 (poisson distribution). The two trains are
independent.
What is the probability that the first international train
arrives within 3 times the arrival time of the first national
train?

Suppose the random variable X has Poisson distribution with rate
parameter l. Let g be the function defined by g(u) = 1/(u+1) , u
>0. Show E(g(X)) >g(E(X)).

A fair coin has been tossed four times. Let X be the number of
heads minus the number of tails (out of four tosses). Find the
probability mass function of X. Sketch the graph of the probability
mass function and the distribution function, Find E[X] and
Var(X).

Suppose that the number of eggs that a hen lays follows the
Poisson distribution with parameter λ = 2. Assume further that each
of the eggs hatches with probability p = 0.8, and different eggs
hatch independently. Let X denote the total number of survivors.
(i) What is the distribution of X? (ii) What is the probability
that there is an even number of survivors? 1 (iii) Compute the
probability mass function of the random variable sin(πX/2) and its
expectation.

Let X be a random variable with probability density function
fX(x) = {c(1−x^2)if −1< x <1, 0 otherwise}.
a) What is the value of c?
b) What is the cumulative distribution function of X?
c) Compute E(X) and Var(X).

Topic: Stochastic processes and poisson processes
Let W1,W2, ... be the event times in a Poisson
process {X(t); t > 0} of rate 2. Suppose it is known
that X(1) = n. For k<n,
what is the conditional density function of W1,....Wk-1,.....,Wn
given that Wk=w
Please follow the comments and do the review before you do this
question

You have observed that the number of hits to your web site
follows a Poisson distribution at a rate of 3 per hour. Let X is
the time between hits and it follows Exponential distribution.
1. What is an average time in minutes between two hits?
2. What is the probability, that you will need to wait less than
40 minutes between two hits?
3. What is the probability, that there will be 2 hits in the
next hour?
1....

Question 1: Compute the moment generating
function M(t) for a Poisson random variable.
a) Use M’(t) to compute E(X)
b) Use M’’(t) to compute Var(X)

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