1. Which of the following are valid probability density functions? Justify your answer.

a. f( x) = (.2ix(.6)1-x 110,l)(x)

b. f( x) = (.3J(.7Jx110,1,2,...i(x}

c. f(x) = .6cx/4J10,001(x)

d. f(x) = x-1l[!,ej(X)

2. Graph each of the density functions in Problem 1.

3. Sparkle Cola, Inc., manufactures a cola drink. The cola is sold in 12 oz. bottles. The probability distribu­ tion associated with the random variable whose out­ come represents the actual quantity of soda placed in a bottle of Sparkle Cola by the soda bottling line is spec­ ified to be f( x) = 50[e-100(12-xlJ 1-oo,121(xJ + e-1001x-12JJ112,ooi(x)].

In order to be considered full, a bottle must contain within .25 oz. of 12 oz. of soda.

a. Define a random variable whose outcome indicates whether or not a bottle is considered full.

b. What is the range of this random variable?

c. Define a probability density function for the ran­ dom variable. Use it to assign probability to the event that a bottle is "considered full."

d. The PDF f(x) is only an approximation. Why?