Question

In the table below the random variable X represents the number of people waiting in line at a fast food restaurant during the lunch hour.

x

0 1 2 3 4 5 Probability 0.0110.035 0.285 0.304 0.239 0.126

a) Verify that this is a discrete probability distribution. b) Draw a probability histogram. c) Compute and interpret the mean of the random variable X. d) Compute the standard deviation of the random variable X. e) What is the probability that there are two people waiting in line for lunch? f) What is the probability that there are more than three people waiting in line for lunch?

Answer 1

(a) The sum of probabilities for all possible events given = 0.011 + 0.035 + 0.285 + 0.304 + 0.239 + 0.126 = 1 So t is indeed a valid discrete probability distribution

(b) We construct a probability histogram by plotting the values taken by random variable on the x axis and their probabilities are represented by the heights of bars as follows:

(c) Mean = E(X) = 0.P(X = 0) +  1.P(X = 1) + 2.P(X = 2) + 3.P(X = 3) + 4.P(X = 4) + 5.P(X = 5)

= 0 + 0.035 + 0.570 + 0.912 + 0.956 + 0.630 =  3.103

It can be intepreted as that on an average there are about 3 people waiting in line at a fast food restaurant during lunch hour.

(d) E(X^2) = 0.P(X = 0) + 1.P(X = 1) + 4.P(X = 2) + 9.P(X = 3) + 16.P(X = 4) + 25.P(X = 5)

= 0 + 0.035 + 1.140 + 2.736 + 3.824 + 3.150 = 10.885

So Var (X) = E(X^2) - E(X)^2 =  10.885 - 3.103^2 = 1.256391

So SD = 1.4082742069

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