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Engineering Probability and Statistics - SE-205 -Chap 4 By S. O. Duffuaa Loading Introduction to Probability Density Function y x (1) F ( x) x Density function of loading on a long, thin beam f(x) Introduction to Probability Density Function y x P(a < X < b) a (1) F ( x) b Density function of loading on a long, thin beam x Probability Density Function For a continuous random variable X, a probability density function is a function such that (1) f ( x) 0 (2) f ( x)dx 1 b (3) P(a X b) f ( x)dx area under f ( x) from a to b for any a and b a Probability for Continuous Random Variable If X is a continuous variable, then for any x1 and x2, P( x1 X x2) P( x1 X x2 ) P( x1 X x2 ) P( x1 X x2 ) 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, P( X 12.60) f ( x)dx 12.6 20( x 12.5 ) 20( x 12.5 ) 20 e dx e |12.6 0.135 12.6 What proportion of parts is between 12.5 and 12.6 millimeters ? Now, 12.6 P(12.5 X 12.6) .6 f ( x)dx e 20( x 12.5) |12 12.5 0.865 12.5 Because the total area under f(x) equals one, we can also calculate P(12.5<X<12.6) = 1 – P(X>12.6) = 1 – 0.135 = 0.865 Cumulative Distribution Function The cumulative distribution function of a continuous random variable X is x F ( x) P( X x) f (u)du for x 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 x F ( x) f (u )du 0.05 x , Finally, for 0 x 20 0 x F ( x) f (u )du 1, for 20 x 0 Therefore, 0 F ( x) 0.05 x 1 The plot of F(x) is shown in Fig. 5-6 x0 0 x 20 20 x 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 E ( X ) xf ( x)dx The variance of X, denoted as V(X) or 2, is 2 V ( X ) ( x ) 2 f ( x)dx x 2 f ( x)dx 2 The standard deviation of X is = [V(X)]1/2 Uniform Distribution A continuous random variable X with probability density function f ( x) 1 /(b a) , has a continuous uniform distribution a xb Uniform Distribution The mean and variance of a continuous uniform random variable X over a x b are E ( X ) (a b) / 2 and 2 V ( X ) (b a) 2 / 12 Applications: • Generating random sample • Generating random variable Normal Distribution A random variable X with probability density function 1 f ( x) e 2 ( x )2 2 2 for x has a normal distribution with parameters , where - < < , and > 0. Also, E ( X ) and V ( X ) 2 Normal Distribution P( X ) 0.6827 P( 2 X 2 ) 0.9545 P( 3 X 3 ) 0.9973 f(x) - 3 - 2 - - - 2 68% 95% 99.7% Probabilities associated with normal distribution - 3 x 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 ( z ) P( Z z ) Standardization If X is a normal random variable with E(X) = and V(X) = 2, then the random variable Z X is a normal random variable with E(Z) = 0 and V(Z) = 1. That is , Z is a standard normal random variable. Standardization Suppose X is a normal random variable with mean and variance 2 . Then, X x P ( X x ) P P( Z z ) 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 Binomial Approximation If X is a binomial random variable, then X np Z np(1 p) is approximately a standard normal random variable. The approximation is good for np > 5 and n(1-p) > 5 Poisson Approximation If X is a Poisson random variable with E(X) = and V(X) = , then Z X is approximately a standard normal random variable. The approximation is good for >5 Do not forget correction for continuity 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 f ( x) ex , for 0 x If the random variable X has an exponential distribution with parameter , then E(X) = 1/ and V(X) = 1/ 2 Lack of Memory Property For an exponential random variable X, P( X t1 t2 | X t1 ) P( X t2 ) Applications: • Models random time between failures • Models inter-arrival times between customers 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 f ( x) r x r 1e x (r 1)! , for x 0 and r 1,2, ... 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 Gamma Function The gamma function is (r ) x r 1e x dx, 0 for r 0 Gamma Distribution The random variable X with probability density function f ( x) r x r 1e x ( r ) , for x 0 has a gamma distribution with parameters > 0 and r > 0. If r is an integer, then X has an Erlang distribution. 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 Weibull Distribution The random variable X with probability density function f ( x) x 1 e ( x / ) , for x 0 has a Weibull distribution with scale parameters > 0 and shape parameter > 0 Applications: • Time to failure for mechanical systems • Time to complete a task. Weibull Distribution If X has a Weibull distribution with parameters and , then the cumulative distribution function of X is F ( x) 1 e x If X has a Weibull distribution with parameters and , then the mean and variance of x are 1 1 and 1 2 2 1 1 2 2 2