Classnotes_7 - 2 Continuous Random Variables 3 Probability...

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Unformatted text preview: 2 Continuous Random Variables 3 Probability Distributions and Probability Density Functions 4 Probability Density Function y y Defined 5 Probability Density Functions and Histograms 6 A Probability Density Function Example Example 4-1 E l 41 Let the continuous random variable X denote the current measured in a thin copper wire p g in milliamperes. Assume that the range of X is [0, 20 mA], and assume that the probability density function of X is f(x) = 0.05 for 0 ” x ” 20. What is the probability that a current measurement is less than 10 milliamperes? The Th probability density function is shown in the figure below (it is assumed that f( ) = b bili d i f i i h i h fi b l (i i d h f(x) 0 wherever it is not specifically defined). The probability requested is indicated by the shaded area in the figure, which can be computed using the equation from slide #5.. 7 Another Probability Density y y Function Example Example 4-2 p 8 The Cumulative Distribution Function in the Continuous Case 9 A Cumulative Distribution Function Example Example 4-4 p 10 Cumulative Distribution Function Examples • Here is the cdf for Example 4-2: • Here is the cdf for Example 4-1: 11 Mean and Variance of a Continuous Random Variable 12 Example of Mean and Variance of a Continuous Random Variable Example 4-6 13 The Mean of a Function of a Continuous Random Variable 14 The Continuous Uniform Distribution 15 The Mean and Variance of the Continuous Uniform Distribution • Use these form la for the situation in formula sit ation example 4-1 (and 4-6) to verify the results from slide #13 that when a=0 and b=20: ȝ = 10.0 and ı2 = 33.33 16 The cdf of the General Continuous Uniform Distribution 17 The Normal Distribution 18 Example Normal Distribution pdfs 19 Well-Known Normal Distribution Probabilities • F any normal random variable: For l d i bl 20 The Standard Normal Distribution 21 How to Use a Table of the cdf of the Standard Normal Distribution Example 4-11 4 11 Figure 4-13 Standard normal probability density function. 22 Using Th St d d N U i The Standard Normal cdf to Determine l df t D t i Probabilities for Other Normal Distributions 23 An Example on Standardizing a p g Normal Random Variable Example 4-13 4 13 24 Another Example (4-14) on Standardizing A p g Normal Random Variable 25 The Normal Approximation to the Binomial Distribution 26 An Example ( p (4-17 & 4-18) of a Normal ) Approximation to the Binomial 27 Guidelines for Using the Normal to Approximate the Binomial or pp Hypergeometric 28 The Normal Approximation to the Poisson Distribution 29 An Example of a Normal Approximation to the Poisson Example 4-20 30 The Exponential Distribution The exponential distribution cdf is: O F(x) = 1-e-Ox 31 An Exponential Distribution Example (Example 4-21) 32 An Exponential Distribution Example (Example 4 21 continued) 4-21 33 An Exponential Distribution p Example (Example 4-21 continued) 34 An Unusual Property of the Exponential Distribution Lack of Memory Property E.g. In Example 4-21, suppose that there are no log-ons from 12:00 to 12:15; the probability that there are no log-ons from 12:15 to 12:21 is still 0 082 Because we have already been 0.082. waiting for 15 minutes, we feel that we are “due.” That is, the probability of a log-on in the next 6 minutes should be greater than 0.082. H t th 0 082 However, f an exponential for ti l distribution this is not true. 35 The Erlang Distribution The random variable X that equals the interval length until r h d i bl h l h i ll h il counts occur in a Poisson process with mean Ȝ > 0 has and Erlang random variable with parameters Ȝ and r The r. probability density function of X is for f x > 0 and r =1, 2, 3, …. d 36 Example pdfs for the Erlang Distribution Misc. Erlang pdfs for different values of r (the "order") {O=0.5} 1.0 10 0.9 0.8 0.7 0.6 0.5 05 0.4 0.3 0.2 0.1 0.0 0.0 1.0 2.0 r=1 3.0 r=2 4.0 r=4 5.0 6.0 7.0 8.0 r=8 37 The Gamma Distribution 38 Example pdfs for the Gamma Distribution Figure 4-25 Gamma probability density functions for selected values of r and O. 39 The Weibull Distribution 40 Example pdfs for the Weibull Distribution Figure 4-26 W ib ll Fi 4 26 Weibull probability density functions for selected values of D and E. 41 The Lognormal Distribution 42 Example pdfs for the Lognormal Distribution 43 44 ...
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