LECTURE UNIT 4.3 Normal Random Variables and Normal Probability Distributions

Understanding Normal Distributions is Essential for the Successful Completion of this Course

Recall: Probability Distributions p(x) for a Discrete Random Variable H p(x) = Pr(X=x) H Two properties 1. 0  p(x)  1 for all values of x 2.  all x p(x) = 1

Graph of p(x); x binomial n=10 p=.5; p(0)+p(1)+ … +p(10)=1 Think of p(x) as the area of rectangle above x p(5)=.246 is the area of the rectangle above 5 The sum of all the areas is 1

Recall: Continuous r. v. x H A continuous random variable can assume any value in an interval of the real line (test: no nearest neighbor to a particular value)

Discrete rv: prob dist function Cont. rv: density function H Discrete random variable p(x): probability distribution function for a discrete random variable x H Continuous random variable f(x): probability density function of a continuous random variable x

Binomial rv n=100 p=.5

The graph of f(x) is a smooth curve f(x)

Graphs of probability density functions f(x) H Probability density functions come in many shapes H The shape depends on the probability distribution of the continuous random variable that the density function represents

Graphs of probability density functions f(x) f(x)

a b Probabilities: area under graph of f(x) P(a < X < b) = area under the density curve between a and b. P(X=a) = 0 P(a < x < b) = P(a < x < b) f(x) P(a < X < b) X

Properties of a probability density function f(x) H f(x)  0 for all x H the total area under the graph of f(x) = 1 H 0  p(x)  1 H  p(x)=1 Think of p(x) as the area of rectangle above x The sum of all the areas is 1 x Total area under curve =1 f(x)

Important difference H 1. 0  p(x)  1 for all values of x 2.  all x p(x) = 1 H values of p(x) for a discrete rv X are probabilities: p(x) = Pr(X=x); H 1. f(x)  0 for all x 2. the total area under the graph of f(x) = 1 H values of f(x) are not probabilities - it is areas under the graph of f(x) that are probabilities

Next: normal random variables