Question

There are two traffic lights on a commuter's route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning home from work. Suppose these two variables are independent, each with pmf given in the accompanying table. (so X1, X2 is a random sample size n=2.)

x1 = 0 1 2

p(x1)= .2 .5 .3

gamma=1.1 sigma^2=.49

a. Determine the pmf of To=X1+X2

b. Calculate gamma_To. How does it relate to gamma, the population mean?

c. Calculate sigma^2_To. How does it relate to sigma^2, the population variance?

d. Let X3 and X4 be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the firsit day. With To= the sum of all four Xi's, what now are the values of E(To) and V(To)?

e. Referring back to (d), what are the values of P(To=8) and P(To>=7)? [Hint: dont even think of listing all the possible outcomes?]

Answer 1

Concepts and reason

A variable which can assume only countable number of real values and for which the value of the variable takes depends on chance is known as a discrete random variable.

If the random variable X is a one-dimensional discrete random variable taking at most a countable infinite number of values then its probabilistic behaviour at each real point is described by a function called the probability mass function.

It is denoted as,

The probability mass function of a discrete random variable X is a function that should satisfy the following properties.

Fundamentals

The expected value of the random variable can be defined as,

And

The variance of the random variable is,

a.

There are two traffic lights on the way to work. Let be the number of lights at which commuter must stop and suppose that the distribution of is as follows:

Given the mean and variance of population are,

Let be the number of lights at which commuter must stop on the way home and it is independent of. Assume that has the same distribution as so that is a random sample of size.

Let be the total number of traffic signal lights in the way to and from work that is.

The probabilities of total number of lights are calculated as follows:

b.

The mean of the probability mass function is,

As the sample size, then is true since is twice that of the sample mean.

c.

The variance of the probability mass function is,

As the sample size,

d.

The number of lights required when driving for 2 independent days is 4. is the sum of all four that is .

The mean of total number of lights is,

The variance of total number of lights is,

e.

The probability that the commuter stops 8 times that is at all the signal lights to and from the work is,

The probability that the commuter stops at least 7 times at signal lights is,

Ans: Part a

The probability mass function of is,

Part b

The mean of sampling distribution is 2.2 and it is twice of the population mean.

Part c

The variance of sampling distribution is 0.98 and it is twice of the population variance.

Part d

The mean and variance of total number of lights are 4.4 and 1.96 respectively.

Part e

The probability values are and.