Presentation on theme: "Probability Distributions"— Presentation transcript:
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 valuesA 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 stopThe lifetime of a transistorA randomly selected number between 0 and 1Let R be a future value of a weekly ratio of closing prices for IBM stockLet W be the exact weight of a randomly selected student
3 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 numbersRange: fX(x)≥0The area under the entire curve is 1Such 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]AabfX
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(Xx)0 FX(x) 1, for all xDomain is all real numbers
8 ExampleThe 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 ExampleThe c.d.f. of T (for the previous example) is given below.Find
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) X=x fX(x) x fX(x) fX(0) 0 fX(0) 1 fX(1) 1 fX(1) 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 0So when we try to compute the expected value we’re adding up a whole bunch of zerosX=xfX(x)x fX(x)fX(0)0 fX(0)1fX(1)1 fX(1)2fX(2)2 fX(2)
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 115bBasically we need to find the area under the entire curve for the function x fX(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 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.XfX
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 = 1Note: The mean does not divide the area in half!x-coordinate that does this is the median.In previous example, the median is approximately
20 Special DistributionA 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.
23 Questions: Find Find E(X) Notice that The expected value of W is given by:.
24 Special DistributionAn 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 ExampleOn 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.
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 webpageMake sure you are able to distinguish between the various types of distributions, their expected values, the c.d.f. and p.d.f.