AGENDA Continuous Random Variables Normal Distribution Normal Approximation to the Binomial Uniform Distribution Lognormal Distribution Gamma Distribution Exponential Distribution NORMAL DISTRIBUTION Most important distribution Frequently encountered in both Nature Man made processes AKA Bell curve Gaussian distribution Particular properties Symetrical about the mean 68% of observations are found +- 1 std from mean 95% of observations are found +- 2 std from mean 99.7% of observations are found +-3 std from mean NORMAL DISTRIBUTION NORMAL DISTRIBUTION NORMAL DISTRIBUTION NORMAL DISTRIBUTION STANDARD NORMAL DISTRIBUTION STANDARD NORMAL DISTRIBUTION USING THE STANDARD NORMAL TABLES The Norm (0,1) is tabulated (table 3) The z value is on the perimeter Cumulative Probability to the left of the z value is in the table CALCULATING SOME STANDARD NORMAL PROBABILITIES Probability of having a standard normal random variable Between 0.87 and 1.28 Look in the Z table F(1.28)=0.8997 F(0.87)=0.8078 F(1.28)-F(0.87)=0.8997-0.8078=0.0919 CONVERTING NORMAL DISTRIBUTION TO STANDARD NORMAL X is a normally distributed random variable with mean and standard deviation The Z table can now be used with other mean and standard deviation values CALCULATING SOME NORMAL PROBABILITIES Normally distributed radiation exposure with a mean of 4.35 mrem and standard deviation of 0.59 mrem. Probability of being exposed to between 4 and 5 mrem. UNIFORM DISTRIBUTION All observations between the minimum and maximum specified values are equally likely Parameters Alpha=minimum value Beta=maximum value UNIFORM DISTRIBUTION UNIFORM DISTRIBUTION LOGNORMAL DISTRIBUTION Random variable whose logarithm is normally distributed Parameters Alpha Beta These don’t have meaning in the same sense as the uniform distribution Their values determine the shape of the distribution LOGNORMAL DISTRIBUTION LOGNORMAL DISTRIBUTION GAMMA DISTRIBUTION GAMMA DISTRIBUTION GAMMA DISTRIBUTION EXPONENTIAL DISTRIBUTION Gamma distribution with alpha parameter=1 Related to random poisson processes Poisson distribution was number of observations in a given period of time Exponential distribution is the time between observations for a poisson process EXPONENTIAL DISTRIBUTION EXPONENTIAL DISTRIBUTION EXPONENTIAL DISTRIBUTION The exponential distribution function can be integrated and evaluated to obtain cumulative distribution function values EXAMPLE Trucks arrive randomly to a wharehouse at an average rate of 3 per hour What is the probability that the interarrival time between trucks will be less than 5 minutes? Solution Integrate the probability function using e^u du substitution Evaluate between 0 and 1/12 hour 1-e^(-1/4)=0.221