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

2 Dr. Jerrell T. Stracener, SAE Fellow EMIS 7370 STAT 5340 Probability and Statistics for Scientists and Engineers Leadership in Engineering Department of Engineering Management, Information and Systems Continuous Probability Distributions Continuous Random Variables & Probability Distributions

3 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 X or Y, are used to denote random variables. Corresponding lower case letters, x or y, are used to denote particular values of the random variables X or Y. Random Variable

4 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

5 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. Continuous Random Variable

6 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) Continuous Random Variable

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

8 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 P(a < X < b) = Probability Density Function

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

10 Probability Density and Distribution Functions f(x) = Probability Density Function x Area = P(x 1 < <x 2 ) F(x) = Probability Distribution Function x F(x 2 ) F(x 1 ) x2x2 x1x1 P(x 1 < <x 2 ) = F(x 2 ) - F(x 1 ) 1 cumulative area x2x2 x1x1

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

12 Variance of X: Standard Deviation of : Mean & Standard Deviation of a Continuous Random Variable X

13 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

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

15 If the probability density function of X is for 0 < x < 1 elsewhere then find (a)  and  (b) P(X>0.4) (c) the value of x* for which P(X<x*)=0.90 Example

16 First, plot f(x): Example

17 Find the mean and standard deviation of X, Example Solution

18 Example Solution

19 and the standard deviation is Example Solution

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

21 Uniform Distribution

22 Probability Density Function ab 0 f(x) x 1/(b-a) Uniform Distribution

23 Probability Distribution Function ab 0 F(x) x 1 Uniform Distribution

24 Mean  = (a+b)/2 Standard Deviation Uniform Distribution

25 Example – Uniform Distribution

26 Example Solution – Uniform Distribution