1.A certain type of component is packaged in lots of four. Let X represent the n

Question

1.A certain type of component is packaged in lots of four. Let X represent the number of properly functioning components in a randomly chosen lot. Assume that the probability mass function of X is given by

Find the value of the constant c so that is a probability mass function. You must show all of your work in order to receive full credit. (2 points)

2.A survey of cars on a certain street of highway during morning commute hours showed that 70% had only one occupant, 15% had 2, 10% had 3, 3% had 4, and 2% had 5. Let X represent the number of occupants in a randomly chosen car. Find the probability mass function of X. (5 points)

Explanation / Answer

Question 1:

For this to be a valid probability density function, the sum of probabilities for all x should be equal to 1. Therefore we get:

f(1) + f(2) + f(3) + f(4) = 1

c + 2c + 3c + 4c = 1

10c = 1

c = 0.1

Therefore the required value of c is 0.1 here.

Question 2:

Here we are given that 70% only had one occupant, therefore P(X=1) = 0.7. Similarly all other probabilities are computed from the given percentages. Therefore the PMF of the given random variable, is represented as:

This is the required PMF of the given random variable X.

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