Download CONTINUOUS RANDOM VARIABLES

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. f ( x)  0, for all x  R.

2.
 f ( x)dx  1.

b
3. P(a  X  b)   f ( x)dx
a
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 bellshaped 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.