Probability Density FunctionCONTINUOUS PROBABILITYDISTRIBUTIONSWe will be concerned with values within an interval.Pr (TC = 190) vs Pr (180  TC  200)Probability of a specific value = 0.Def: Probability Density Function of a random variable, X, is a function such that the area under the curve corresponding to the functionbetween any two points, a, b, is equal to the probability that X falls between a and b. Areaunder the curve for all possible values equals 1 or 100%. This represents the distribution ofa continuous Random Variable. We will be interested in the relative frequency of the occurrence of values between 2 points on the X-axis, say a and b. We want the area in that segment.Def: Cumulative Density Function for a Random Variable, X, evaluated at a point, a, is the probability that X  a. It is represented by thearea under the Probability Density Function to the left of a. Note: Pr (X  a) and Pr (X < a) are the same for a continuous Random Variable. This is not the case for a discrete Random Variable. Note: Expected value = E(X) =  is the average value for the random variable. Note: Variance = Var(X) = 2 = E(X - )2 Standard Deviation =  2XVarNORMAL DISTRIBUTION (GAUSSIANDISTRIBUTION)Probability Density Functionxwhereeσπf(x)μ)(Xσ222121,  are two parameters,  > 0e = 2.71828 = 3.14159 determines the location or center of the distribution2 determines the spreadCharacteristics:1. Symmetric about the mean, 2. Mean = median = mode3. Area under the curve = 1 implies 50% of the area to the left of the mean and 50% to the right of the mean4. About 68% of the area is between  - 1 and  + 15. About 95% of the area is between  - 1.96 and  + 1.966. For every  and , there is a different Gaussian DistributionNotation: N(,2)σπHeight21Height is inversely proportion to . The greater 2, the lower the peak. If 21 < 22, then the height of N(,21) > N(,22).STANDARD NORMAL DISTRIBUTION—N(0, 1)xwhereeπf(x))x(22121Characteristics:1. Symmetric about X = 0, f(x) = f(-x)2. Pr(-1 < X < +1) = 0.68273. Pr(-1.96 < X < 1.96) = 0.954. Pr(-2.576 < X < 2.576) = 0.995. Pr(-3.29 < X < 3.29) = 0.999The Standard Normal Distribution is very well tabled. The tables are based on the Cumulative Distribution Function,(X) = Pr (X  x) where X ~ N(0, 1).Tables are on pages 818-821.Column A – Area from - to x. This is the Pr(X x).Column B = Area from X to +. The is the Pr(X x).Examples: If X ~ N(0, 1), find1. Pr (X  1.64)2. Pr (X  0.50)3. Pr (X  0)4. Pr(X  -0.50)From Column A, Pr(X  -0.50) = Pr(X  0.50)= 1- Pr(X  0.50)= 1 - 0.6915= 0.3085From Column B, read directly Pr(X  0.50).5. Pr (X  -1.2)6. Pr (-1.3  X 1.75)Column D – Area symmetric about 0. Pr (-x  X  x)7. Pr (-2  X  2)8. Pr (-1.2  X  1.2)Column C – Area from 0 to a point. Pr (0  X  x)9. Pr (0  X  1.6)Note: Pr (-1.3  X  0) is the same as Pr (0  X  1.3)We can also use the tables in reverse. What is the value of X such that 60% of the area is below that value? Go to Column A and look for 0.60 in the column. Read the value of x. This is sometimes called the inverse normal function. This is also sometimes called percentiles.Pr(X < zu) = u, where X ~ N(0, 1)Find,1. z.952. z.9753. z.994.