1. Engineering Probability and Statistics - SE-205 -Chap 4 By S. O. Duffuaa

2. Introduction to Probability Density Function Density function of loading on a long, thin beam x Loading

3. Introduction to Probability Density Function Density function of loading on a long, thin beam x f(x) a b P(a < X < b)

4. Probability Density Function For a continuous random variable X, a probability density function is a function such that

5. Probability for Continuous Random Variable If X is a continuous variable, then for any x 1 and x 2,

6. Example Let the continuous random variable X denote the diameter of a hole drilled in a sheet metal component. The target diameter is 12.5 millimeters. Most random disturbances to the process result in larger diameters. Historical data show that the distribution of X can be modified by a probability density function f(x) = 20e -20(x-12.5), x  12.5. If a part with a diameter larger than 12.60 millimeters is scrapped, what proportion of parts is scrapped ? A part is scrapped if X  12.60. Now, What proportion of parts is between 12.5 and 12.6 millimeters ? Now, Because the total area under f(x) equals one, we can also calculate P(12.5 12.6) = 1 – 0.135 = 0.865

7. Cumulative Distribution Function The cumulative distribution function of a continuous random variable X is

8. Example for Cumulative Distribution Function For the copper current measurement in Example 5-1, the cumulative distribution function of the random variable X consists of three expressions. If x < 0, then f(x) = 0. Therefore, F(x) = 0, for x < 0 Finally, Therefore, The plot of F(x) is shown in Fig. 5-6

9. Mean and Variance for Continuous Random Variable Suppose X is a continuous random variable with probability density function f(x). The mean or expected value of X, denoted as  or E(X), is The variance of X, denoted as V(X) or  2, is The standard deviation of X is  = [V(X)] 1/2

10. Uniform Distribution A continuous random variable X with probability density function has a continuous uniform distribution

11. Uniform Distribution The mean and variance of a continuous uniform random variable X over a  x  b are Applications: Generating random sample Generating random variable

12. Normal Distribution A random variable X with probability density function has a normal distribution with parameters , where -  0. Also,

13. Normal Distribution 68%  - 3   - 2   -   -   - 2   - 3  x 95% 99.7% f(x)f(x) Probabilities associated with normal distribution

14. Standard Normal A normal random variable with  = 0 and  2 = 1 is called a standard normal random variable. A standard normal random variable is denoted as Z. The cumulative distribution function of a standard normal random variable is denoted as

15. Standardization If X is a normal random variable with E(X) =  and V(X) =  2, then the random variable is a normal random variable with E(Z) = 0 and V(Z) = 1. That is, Z is a standard normal random variable.

16. Standardization Suppose X is a normal random variable with mean  and variance  2. Then, where, Z is a standard normal random variable, and z = (x -  )/  is the z-value obtained by standardizing X. The probability is obtained by entering Appendix Table II with z = (x -  )/ . Applications: Modeling errors Modeling grades Modeling averages

17. Binomial Approximation If X is a binomial random variable, then is approximately a standard normal random variable. The approximation is good for np > 5 and n(1-p) > 5

18. Poisson Approximation If X is a Poisson random variable with E(X) = and V(X) =, then is approximately a standard normal random variable. The approximation is good for > 5 Do not forget correction for continuity

19. Exponential Distribution The random variable X that equals the distance between successive counts of a Poisson process with mean > 0 has an exponential distribution with parameter. The probability density function of X is If the random variable X has an exponential distribution with parameter, then E(X) = 1/ and V(X) = 1/ 2

20. Lack of Memory Property For an exponential random variable X, Applications: Models random time between failures Models inter-arrival times between customers

21. Erlang Distribution The random variable X that equals the interval length until r failures occur in a Poisson process with mean > 0 has an Erlang distribution with parameters and r. The probability density function of X is

22. Erlang Distribution If X is an Erlang random variable with parameters and r, then the mean and variance of X are  = E(X) = r/ and  2 = V(X) = r/ 2 Applications: Models natural phenomena such as rainfall. Time to complete a task

23. Gamma Function The gamma function is

24. Gamma Distribution The random variable X with probability density function has a gamma distribution with parameters > 0 and r > 0. If r is an integer, then X has an Erlang distribution.

25. Gamma Distribution If X is a gamma random variable with parameters and r, then the mean and variance of X are  = E(X) = r/ and  2 = V(X) = r/ 2 Applications: Models natural phenomena such as rainfall. Time to complete a task

26. Weibull Distribution The random variable X with probability density function has a Weibull distribution with scale parameters  > 0 and shape parameter  > 0 Applications: Time to failure for mechanical systems Time to complete a task.

27. Weibull Distribution If X has a Weibull distribution with parameters  and , then the cumulative distribution function of X is If X has a Weibull distribution with parameters  and , then the mean and variance of x are and