1. Continuous Random Variables Much of the material contained in this presentation can be found at this excellent website http://people.hofstra.edu/faculty/Stefan_ Waner/cprob/cprob2.html

2. Suppose that you have purchased stock in Google and each day you note the closing price of the stock. The result each day is a real number X (the closing price of the stock) in the unbounded interval [0, + infinity). OR Suppose that you time several people running a 50-meter dash. The result for each runner is a real number X, the race time in seconds. In both cases, the value of X is somewhat random. What are they?

3. In other words, X can take on essentially any real value in some interval, rather than, say, just integer values. For this reason we refer to X as a continuous random variable. SO to define it fully Continuous Random Variable A random variable is a function X that assigns to each possible outcome in an experiment a real number. If X may assume any value in some given interval I, it is called a continuous random variable. If it can assume only a number of separated values, it is called a discrete random variable.

4. If X is a random variable, we are usually interested in the probability that X takes on a value in a certain range. For instance, if X is the daily closing price of Google stock and we find that 60% of the time the price is between $10 and $20, we would say The probability that X is between $10 and $20 is 0.6. We write this statement mathematically as follows. What does it all mean?

5. Age of a Rental Car Years0-11-22-33-44-55-66-7 Prob.0.20.280.20.150.10.050.02 Histogram (sum of areas = ……) This is only true if the width of the bar is ………. A histogram can be drawn

6. A histogram is very restrictive when dealing with continuous random variables as you may not always want to only consider the probability between standard intervals such as Instead you may want to find Therefore we can approximate a curve to match the pattern of the distribution y = f(x) This is called a Probability Density Function

7. Now that we have a function defined, y = f(x) and we know from the histogram that area can define the probability for certain intervals We can now make use of our skills in Calculus to find the area under the curve we want between our specified intervals. This will give us the Probability we seek. Hmm, I think I can see where this is heading Curves, Area CALCULUS!!!

8. Probability Density Function A probability density function (or probability distribution function) is a function f defined on an interval (a, b) and having the following properties. (a) for every x (b)

9. Putting it all together Suppose now that as in the previous section, we wanted to calculate the probability that a rented car is between 0 and 4 years old. Referring to the table, = 0.20 + 0.28 + 0.20 + 0.15 = 0.83 Referring to the following figure, notice that we can obtain the same result by adding the areas of the corresponding bars, since each bar has a width of 1 unit. Ideally, our probability density curve should have the property that the area under it for 0 X 4 is the same, that is,

10. Probability Associated with a Continuous Random Variable A continuous random variable X is specified by a probability density function f. The probability is specified by

11. ENOUGH! LET ME HAVE A TRY! For what constant k is f(x) = ke -x a probability density function on [0,1]? Hint: We need to choose a k that makes requirements (a) and (b) of the definition true. Since e x > 0 for all x, all we need for (a) is to make sure that we choose k 0. For (b), we calculate ???? Answer: k = e/(e-1)