Probability Distributions

1. Probability Distributions Continuous Random Variables

2. Continuous Random Variables • A random variable was a numerical value associated with the outcome of an experiment. • Finite discrete random variables were ones in which the values were countable whole numbered values • A continuous random variable is a random variable that can assume any value in some interval of numbers, and are thus NOT countable. • Examples: • The time that a train arrives at a specified stop • The lifetime of a transistor • A randomly selected number between 0 and 1 • Let R be a future value of a weekly ratio of closing prices for IBM stock • Let W be the exact weight of a randomly selected student

3. fX A a b Continuous Random Variables • A random variable is said to be continuous if there is a function fX(x) with the following properties: • Domain: all real numbers • Range: fX(x)≥0 • The area under the entire curve is 1 • Such a function fX(x) is called the probability density function (abbreviated p.d.f.) • The fact that the total area under the curve fX(x) is 1 for all X values of the random variable tells us that all probabilities are expressed in terms of the area under the curve of this function. • Example: If X are values on the interval from [a,b], then the P(a≤X≤b) = area under the graph of fX(x) over the interval [a,b]

4. Continuous Random Variables • Because all probabilities for a continuous random variable are described in terms of the area under the p.d.f. function, the P(X=x) = 0. • Why: the area of the p.d.f. for a single value is zero because the width of the interval is zero! • That is, for any continuous random variable, X, P(X = a) = 0 for every number a. This DOES NOT imply that X cannot take on the value a, it simply means that the probability of that event is 0.

5. Continuous Random Variables • Rather than considering the probability of X taking on a given single value, we look for the probability that X assumes a value in an interval. • Suppose that a and b are real numbers with a < b. Recall that X a is the event that X assumes a value in the interval(, a]. Likewise, a < Xb and b < X are the events that X assumes values in (a, b] and (b, ), respectively. These three events are mutually exclusive and at least one of them must happen. Thus, • P(X a) + P(a < Xb) + P(b < X) = 1. • Since we are interested in the probability that X takes a value in an interval, we will solve for P(a < Xb).

6. Continuous Random Variables • Because X is a continuous random variable, P(X = a) = 0 and P(X = b) = 0. Thus, it makes no difference whether or not we include the end points in an interval.

7. The cumulative distribution function • The same probability information is often given in a different form, called the cumulative distribution function, (c.d.f), FX(x) • FX(x)=P(Xx) • 0 FX(x) 1, for all x • Domain is all real numbers

8. Example • The p.d.f. of T, the weekly CPU time (in hours) used by an accounting firm, is given below.

9. Example (cont) • The graph of the p.d.f. is given below:

10. Example (cont) • is equal to the area between the graph of and the t-axis over the interval.

11. Another Example • The c.d.f. of T (for the previous example) is given below. • Find

12. The graph of the c.d.f.

13. Solution

14. Expected Value—Continuous Random Variable • For discrete finite random variables, the expected value was determined by taking each value of the random variable and multiplying it by the corresponding probability as stated by:

15. Expected Value (cont) • For a continuous random variable, the process is more tricky. • For example, suppose we wanted to know the expected value of X which is defined on the interval [0,2]. • Unfortunately, the P(X=a) for any number is 0 • So when we try to compute the expected value we’re adding up a whole bunch of zeros

16. Expected Value (cont) • We need some other way of looking for the expected value for a continuous random variable. • Unfortunately, this requires calculus which you will learn more about in 115b • Basically we need to find the area under the entire curve for the function xfX(x) • We can’t do this without knowing some calculus so we’ll give a more geometric interpretation of the E(X) for a continuous random variable.

17. fX X Expected Value—Geometric Interpretation • We can use the probability density function to give a geometric interpretation for the mean of a continuous random variable, X. Suppose that we draw the p.d.f. on a thin sheet of metal and cut out the region between the graph of f(X) and the x-axis. If we place a knife edge under a line through  on the x-axis and perpendicular to that axis, then the metal sheet will balance on that edge.

18. A more concrete example • Consider the p.d.f. shown below for a continuous random variable X

19. Expected Value—Geometric Interpretation • Note: The mean does not occur at the highest point of the graph! • x-coordinate of highest point is the mode. • In the previous example, this was at x = 1 • Note: The mean does not divide the area in half! • x-coordinate that does this is the median. • In previous example, the median is approximately 0.3466.

20. Special Distribution • A continuous uniform random variable is a random variable defined on an interval such that every subinterval of having the same length has the same probability. • If X is a continuous uniform random variable on the interval , then

21. Example—Uniform Distribution • A bus arrives at a bus stop every 10 minutes. Let W be the waiting time (in minutes) until the next bus. The p.d.f. and c.d.f. of W are given below.

22. Uniform Distribution: p.d.f. & c.d.f.

23. Questions: • Find • Find E(X) • Notice that The expected value of W is given by: .

24. Special Distribution • An exponential random variable may be used to model the length of time between consecutive occurrences of some event in a fixed unit of space or time. • If X is an exponential random variable with parameter, then

25. Example • On average, three customers per hour use the ATM in a local grocery store. Let T be the time (in minutes) between consecutive customers. The p.d.f. and c.d.f. of T are given below.

26. Exponential Distribution: p.d.f. & c.d.f.

27. Questions: • Find • The expected value cannot be determined from the p.d.f. function without using calculus so we’ll simply tell you that: • The expected value of T is given by:

28. fX(x) fX(x) x x a b Summary of Distributions

29. Summary of Distributions

30. Expected Values • Finite Discrete Random Variables: • Continuous Random Variables: • Uniform Distribution: • Exponential Distribution:

31. Additional Information • See Distribution Study Guide under “Worksheets” link on my webpage • Make sure you are able to distinguish between the various types of distributions, their expected values, the c.d.f. and p.d.f.