AGENDA Continuous Random Variables Normal Distribution Normal Approximation to the Binomial Uniform Distribution Lognormal Distribution CONTINUOUS RANDOM VARIABLES Mean of a probability density Variance of a probability density 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. NORMAL APPROXIMATION TO THE BINOMIAL Normal distribution can approximate the binomial Restrictions n is large p is close to 0.50 (not small enough for poisson approx.) Parameters mean=np Var=np(1-p) Substitute in Z formula Needs a continuity correction Have to adjust discrete values into continuous values NORMAL APPROXIMATION TO THE BINOMIAL EXAMPLE 1 20% of memory chips are defective For a lot of 100 randomly chosen chips At most 15 will be defective? Mean=100 * 0.2 Var=100 * 0.2 * (1-0.2) EXAMPLE 2 20% of memory chips are defective For a lot of 100 randomly chosen chips Exactly 15 will be defective? 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