1. Topic 5: Continuous Random Variables and Probability Distributions CEE 11 Spring 2002 Dr. Amelia Regan These notes draw liberally from the class text, Probability and Statistics for Engineering and the Sciences by Jay L. Devore, Duxbury 1995 (4th edition)

2. Definition n A random variable X is said to be continuous if its set of possible values is an entire interval of numbers -- that is, for some A<B, any number x between A and B is possible. n Let X be a continuous rv. The a probability distribution or probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a <= b, n To be a legitimate pdf, f(x) must satisfy the following two conditions: 1. f(x) >= 0 for all x 2.

3. Class exercise n Which of the following functions is a legitimate pdf? n To be a legitimate pdf, f(x) must satisfy the following two conditions: 1. f(x)>=  0 for all x, 2.

4. Definition Let X be a continuous rv with probability density function (pdf) f(x). The expected value or mean value of X, denoted E(x) or  x is given by: n If X is a continuous rv with pdf f(x) and h(x) is a function of X, then

5. Class exercise n Let X be a continuos rv with the following pmf n Calculate the pmf n Now let h(x) = x 2 +2 Calculate E(h(X)) n Calculate E(X)

6. Variance of a random variable If X is a continuous random variable with mean , then the variance of X n As shown for previously for discrete distributions

7. The Uniform distribution n The uniform distribution is one for which all values in the region for which the distribution is defined are equally likely. A common range is [0,1], though any range is possible. The pdf and cdf of the uniform distribution are the following:

8. The Uniform distribution n Class Exercise (Banks et al., 1995, p.203): A bus arrives every 20 minutes at a specified stop beginning at 6:40 AM and continuing until 8:40 AM. A certain passenger does not know the schedule but arrives randomly (uniformly distributed) between 7:00 AM and 7:30 AM every morning. What is the probability that the passenger waits more than 5 minutes for a bus?

9. The Uniform distribution n E(X) and Var(X) can be obtained by integration: n Class exercise: u Integrate the pmf above to find the expression for E(X) and Var(X).

10. The Normal distribution n The normal distribution is probably the most used of the known probability distributions. The pdf is messy: n In fact, integrating the pdf of the normal distribution to obtain the cdf and the mean and variance are not possible. However, the normal distribution has some properties that make it easy to work with.

11. The Normal distribution When we discuss the normal distribution we must specify its mean  and standard deviation . The normal distribution with   0 and  = 1.0 is known as the standard normal distribution. Its graph is shown below:

12. The Normal distribution n The normal distribution is a family of distributions n Any normally distributed variable can be transformed to the standard normal distribution which has mean 0 and standard deviation 1. The relationship between a normal distributed random variable X with mean  and standard deviation  is the following to a standard normal variable Z is the following: n We “standardize” by subtracting a constant and dividing the difference by another constant

13. The Normal distribution n What happens to the mean and the standard deviation when we standardize? n Remember that E(aX+b) = aE(X) + b n Remember also that Var(aX+b) = a 2 Var(X) n Therefore the standard deviation is also 1

14. The Normal distribution n If the population distribution of a variable is (approximately) normal then u Roughly 68% of the values are within one standard deviation of the mean u Roughly 95% of the values are within two standard deviations of the mean u Roughly 99.7% of the values are within three standard deviations of the mean

15. The Normal distribution n In order to calculate probabilities associated with normal random variables we use tables (or a built-in function which uses an approximation method to estimate the values). n To use the tables, we must transform our data to the standard normal distribution n For example, if x is approximately normally distributed with mean 34 and standard deviation 3.5, what is the probability that an observed value will be less than 32?

16. Class exercise n Again if x has mean 34 and standard deviation 3.5, calculate the following probabilities: n X > 39 n X < 36 n X < 30

17. The Exponential distribution n The exponential distribution is among the most useful in science and engineering. It is typically used to predict the time between events. Its pdf and cdf are the following: Please note that this notation is slightly different from that of the text, which uses instead of  as the parameter.

18. The Exponential distribution n The mean and variance of the exponential distribution are given below: The reason for the different notation is important -- its because of the relationship between the exponential and Poisson distributions. If the number of events in a time period (often called arrivals) is Poisson distributed with parameter =  t, then time between successive arrivals is exponentially distributed with parameter .

19. The exponential and the Poisson The time between calls to a suicide help line in a busy city are approximately exponentially distributed with mean  0.50 hours u Calculate the probability that no calls will be received in a 1.0 hour period.  Now assume that the number of calls received per hour is approximately Poisson distributed with parameter  = 2 per hour. u Calculate the probability that in a 1.0 hour period zero calls are received.

20. The exponential and the Poisson + You absolutely must understand the relationship between these distributions + If the time between events is approximately exponentially distributed then then number of events in a time period is approximately Poisson distributed