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540201 Statistics for Engineer Week II and Week III: Random Variables and Probability Distribution Random Variables Controllable Variables Output Input Uncontrollable Variables Random Variable : A Numerical variable whose measured value can change from one replicate of the experiment to another 3-2 Random Variables Discrete random variables Continuous random variables 3-3 Probability The chance of “x” A degree of belief A relative frequency between “event frequency” to the “outcome frequency” Histogram of Compressive Strength Histogram of Compressive Strength 0.275 22 0.2125 0.175 14 0.125 0.075 0.0250.0375 17 10 0.05 0.025 2 3 6 4 2 3-4 Continuous Random Variables Cumulative Distribution Function (cdf) x F ( x) P( X x) f (u)du for x Continuous Random Variables Probability Density Function (pdf) b P(a x b) f ( x)dx a when 1) f ( x) 0 2) f ( x)dx 1 Continuous Random Variables Mean and Variance Example 3.5 Continuous Uniform Distribution Continuous Uniform Distribution Continuous Uniform Distribution Continuous Uniform Distribution 3-5.1 Normal Distribution (Gaussian) Normal Distribution Normal Distribution Normal Distribution Normal Distribution Ex 3-11 Ex 3-12 Normal Distribution Normal Distribution t-Distribution When is unknown Small sample size Degree of freedom (k) = n-1 Significant level = t, k t-Distribution Exponential Distribution Exponential Distribution 3-7 Discrete Random Variables • Probability Mass Function (pmf) Discrete Random Variables Cumulative Distribution Function (cdf) Discrete Random Variables Mean and Variance 3-8 Binomial Distribution A Bernoulli Trial Binomial Distribution Binomial Distribution Example 3-28 Bit transmission errors: Binomial Mean and Variance 3-9 Poison Distribution The random variable X that equals the number of events in a Poison process is a Poison random variable with parameter >0, and the probability mass function of X is x f ( x) e x! The mean and variance of X are E ( x) and V ( x) 3-9 Poison Distribution 3-9 Poison Distribution 3-9 Poison Distribution 3-10 Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Binomial 3-10 Normal Approximation to the Binomial and Poisson Distributions 3-10 Normal Approximation to the Binomial and Poisson Distributions 3-10 Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Poisson 3-10 Normal Approximation to the Binomial and Poisson Distributions 3-10 Normal Approximation to the Binomial and Poisson Distributions Q &A