CONTINUOUS RANDOM VARIABLES

AND THEIR PROBABILITY DENSITY

FUNCTIONS(P.D.F.)

REMINDER

• A discrete random variable is one whose possible values either constitute a finite set [e.g. E = {2, 4, 6, 8, 10}] or else can be listed in an infinite sequence [e.g. N = {0, 1, 2, 3, 4, …}]. A random variable whose set of possible values is an entire interval of numbers is not discrete.

CONTINUOUS RANDOM VARIABLES

• A random variable X is said to be continuous if its set of possible values is an entire interval of numbers – that is, if for some a < b, any number x between a and b is possible.

• FOR EXAMPLE: [2,5]; (- 4, 7); [24, 71); (11, 31].

DEFINITION: PROBABILITY DENSITY FUNCTION (P.D.F.)

• The function f(x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, if

PROBABILITY DENSITY FUNCTION FOR A CONTINUOUS RANDOM VARIABLE, X.

1. .,0)( Rxallforxf

2.

.1)( dxxf

3. b

a

dxxfbXaP )()(

EXAMPLES FROM PRACTICE EXERCISES SHEET 6

SOME CONTINUOUS PROBABILITY DISTRIBUTION FUNCTIONS

• UNIFORM PROBABILITY DISTRIBUTION FUNCTION;

• EXPONENTIAL PROBABILITY DISTRIBUTION FUNCTION;

• NORMAL PROBABILITY DISTRIBUTION FUNCTION.

UNIFORM PROBABILITY DISTRIBUTION FUNCTION

• A continuous random variable, r.v. X, is said to have a uniform distribution on the interval [a,b] if the probability density function, p.d.f. of X is

UNIFORM PROBABILITY DENSITY FUNCTION

EXAMPLES FROM PRACTICE EXERCISES SHEET 6

EXPONENTIAL PROBABILITY DENSITY FUNCTION

• The continuous random variable X has an exponential distribution, with parameter >0 if its density function is given by

EXPECTED VALUE, E(X), AND VARIANCE, VAR(X), OF A CONTINUOUS RANDOM VARIABLE, X, EXPONENTIALLY

DISTRIBUTED

EXAMPLES FROM PRACTICE EXERCISES SHEET 6

NORMAL PROBABILITY DISTRIBUTION FUNCTION,

STANDARD NORMAL PROBABILITY DENSITY FUNCTION, N(0,1)

Z – SCORES OR STANDARDIZED SCORES

REMARKS

• Probably the most important continuous distribution is the normal distribution which is characterized by its “bell-shaped” curve. The mean is the middle value of this symmetrical distribution.

• When we are finding probabilities for the normal distribution, it is a good idea first to sketch a bell-shaped curve. Next, we shade in the region for which we are finding the area, i.e., the probability. [Areas and probabilities are equal] Then use a standard normal table to read the probabilities.

REMARKS CONTINUED

• AREA UNDER N(0,1) = 1• PROBABILITY OF A CONTINUOUS RANDOM

VARIABLE, NORMALLY DISTRIBUTED = AREA UNDER THE BELL SHAPED CURVE.

• STANDARD NORMAL TABLES GIVE AREAS OR PROBABILITIES TO THE LEFT OF THE Z –

SCORES AND TO FOUR DECIMAL PLACES.

EMPIRICAL RULE OR THE 68 – 95 – 99.7% RULE

•

EMPIRICAL RULE OR 68 – 95 – 99.7% RULE

• IN A NORMAL MODEL, IT TURNS OUT THAT • 1. 68% OF VALUES FALL WITHIN ONE

STANDARD DEVIATION OF THE MEAN; • 2. 95% OF VALUES FALL WITHIN TWO

STANDARD DEVIATIONS OF THE MEAN; • 3. 99.7% OF VALUES FALL WITHIN THREE

STANDARD DEVIATIONS OF THE MEAN.

MEAN OR EXPECTED VALUE, E(X), VARIANCE, VAR(X), AND STANDARD DEVIATION SD(X),OF A CONTINUOUS RANDOM

VARIABLE X.