Question

The pizza palace offers three sizes of cola. The smallest size sells for $1.99, the medium for $2.49, and the large for $2.89. Thirty percent of the drinks sold are small, 50% are medium, and 20% are large. Create a probability distribution for the random variable price and answer the following questions.

a. Is this a discrete probability distribution? Indicate why or why not?

b. Compute the mean amount charged for a cola.

c. What is the variance in the amount charged for a cola? the standard deviation?

Answer 1

(a)

Let X is a random variable shows the prize of pizza. Following table shows the pdf of X:

X	P(X=x)
1.99	0.3
2.49	0.5
2.89	0.2
Total	1

It is discrete random variable.

(b-c)

Following table shows the calculations for mean and SD:

X	P(X=x)	x*P(X=x)	x^2*P(X=x)
1.99	0.3	0.597	1.18803
2.49	0.5	1.245	3.10005
2.89	0.2	0.578	1.67042
Total	1	2.42	5.9585

The mean amount charged for a cola is

$E(X)=\sum xP(X=x)=2.42$

The variance in the amount charged for a cola is
$Var(X)=\sum x^{2}P(X=x)-\left [ \sum xP(X=x) \right ]^{2}=5.9585-2.42^{2}=0.1021$

The standard deviation is

$\sigma=\sqrt{\sum x^{2}P(X=x)-\left [ \sum xP(X=x) \right ]^{2}}=\sqrt{5.9585-2.42^{2}}=0.3195$