lecture unit 4.3 normal random variables and normal probability distributions

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 p(x) = Pr(X=x) 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 areaof rectangle above x

p(5)=.246 is the areaof the rectangle above 5

The sum of all theareas is 1

Recall: Continuous r. v. x

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 functionCont. rv: density function Discrete random

variable

p(x): probability distribution function for a discrete random variable x

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)

Probability density functions come in many shapes

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)

f(x) f(x)

a b

Probabilities:area undergraph 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

P(a X b) = f(x)dxa

b

Properties of a probability density function f(x)

f(x)0 for all x the total area under the

graph of f(x) = 1

0 p(x) 1 p(x)=1

Think of p(x) as the areaof rectangle above x

The sum of allthe areas is 1

xx

Total areaTotal areaunder curveunder curve

=1=1

f(x)f(x)

Important difference

1. 0 p(x) 1 for all values of x

2. all x p(x) = 1

values of p(x) for a discrete rv X are probabilities: p(x) = Pr(X=x);

1. f(x)0 for all x

2. the total area under the graph of f(x) = 1

values of f(x) are not probabilities - it is areas under the graph of f(x) that are probabilities

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