Question

A coffee dispenser machine fills cups at amounts that are uniformly distributed. Specifically, if we select a small 12 oz coffee, the machine dispenses between 10 and 13 oz. of coffee, following a uniform distribution. a) Is the amount of coffee dispensed by the machine a discrete or continuous random variable? Explain.b) Graph the probability function for P(X), where X is the amount of coffee dispensed when I select a small coffee. c) What is P( X = 12) ? d) Find the probability that my 12 oz. cup will not overflow, i.e. P( X 12e) How large of a coffee cup should I have if I want to be 95% confident that the dispenser won't overfill the cup?

Answer 1

a)

A random variable is a function that associates a real number with each element in the sample space.

The distinction between a discrete and a continuous random variable is the same as the distinction between a discrete and a continuous frequency distribution: only certain results are possible for a discrete random variable, but any of an infinite number of results within a certain range are possible for a continuous random variable. 

If a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers, it is called a discrete sample space. 

If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space.

The amount of coffee dispensed by the machineis a continuous random variable, because its value can be any real number between 10 and 13 ounces, inclusive.

b) The probability density function of the continuous uniform random variable X on the interval [10, 13] is

$$f(x) = \begin{cases} {1 \over 13-10} &\text{for } 10\leq x\leq 13 \\ 0 &\text{for } x<10 \ \text{or } x>13 \end{cases}$$

$$f(x) = \begin{cases} {1 \over 3} &\text{for } 10\leq x\leq 13 \\ 0 &\text{for } x<10 \ \text{or } x>13 \end{cases}$$

c)

$$P(X=12)=0$$

d)

Find the probability that my 12 oz. cup will not overflow

$$P(X\leq12)={12-10 \over 3}={2 \over 3}$$

How large of a coffee cup should I have if I want to be 95% confident that the dispenser would not overfill the cup?

$$P(X\leq k)={k-10 \over 3}=0.95$$

$$k=3(0.95)+10=12.85$$

I take 12.85 oz. coffee cup.