Continuous Probability Distributions Continuous Random Variables & Probability Distributions

1. Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Continuous Probability Distributions Continuous Random Variables & Probability Distributions Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08

2. Random Variable • Definition - A random variable is a mathematical • function that associates a number with every • possible outcome in the sample space S. • Definition - 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 and a random variable defined over this space is called a continuous random variable. • Notation - Capital letters, usually or , are • used to denote random variables. Corresponding • lower case letters, x or y, are used to denote • particular values of the random variables or .

3. For many continuous random variables or (probability functions) there exists a function f, defined for all real numbers x, from which P(A) can for any event A  S, be obtained by integration: Given a probability function P() which may be represented in the form of Continuous Random Variable

4. Continuous Random Variable in terms of some function f, the function f is called the probability density function of the probability function P or of the random variable , and the probability function P is specified by the probability density function f.

5. Continuous Random Variable Probabilities of various events may be obtained from the probability density function as follows: Let A = {x|a < x < b} Then P(A) = P(a < X < b)

6. Continuous Random Variable Therefore = area under the density function curve between x = a and x = b. f(x) Area = P(a < x <b) 0 0 x a b

7. Probability Density Function The function f(x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, if 1. f(x)  0 for all x  R. 2. 3. P(a < X < b) =

8. The cumulative probability distribution function, F(x), of a continuous random variable with density function f(x) is given by Note: Probability Distribution Function

9. Area = P(x1 < <x2) x x1 x2 F(x) = Probability Distribution Function 1 cumulative area F(x2) P(x1 < <x2) = F(x2) - F(x1) F(x1) x x1 x2 Probability Density and Distribution Functions f(x) = Probability Density Function

10. Mean & Standard Deviation of a Continuous Random Variable X • Mean or Expected Value • Remark • Interpretation of the mean or expected value: • The average value of in the long run.

11. Mean & Standard Deviation of a Continuous Random Variable X • Variance of X: • Standard Deviation of :

12. If a and b are constants and if  = E is the mean and 2 = Var is the variance of the random variable , respectively, then and Rules

13. If Y = g(X) is a function of a continuous random variable , then Rules

14. Example If the probability density function of X is for 0 < x < 1 elsewhere • then find • m and s • P(X>0.4) • the value of x* for which P(X<x*)=0.90

15. Example First, plot f(x):

16. Example Solution Find the mean and standard deviation of X,

17. Example Solution

18. and the standard deviation is Example Solution

19. Example Solution (b) (c) for 0<x<1 Since 1.32>1, so

20. Uniform Distribution

21. Uniform Distribution Probability Density Function f(x) 1/(b-a) 0 x a b

22. Uniform Distribution Probability Distribution Function F(x) 1 0 x a b

23. Mean •  = (a+b)/2 • Standard Deviation Uniform Distribution

24. Example – Uniform Distribution

25. Example Solution – Uniform Distribution