26 Jun 2001

Main Points

Definition
A random variable X is a continuous random
variable (crv) if the range of X
is made up of interval(s).
A crv is a theoretical representation of a continuous variable such
as length, mass or time.
Probability Density Function
A crv is specified by its probability density function
(pdf).
If X is a crv with pdf f(x), then
P(a � X �
b) =_{�} 
�^{b}
�_{a} 
f(x) dx. 
If a = b, then
P(X = a) = P(a �
X � a) = 
�^{a}
�_{a} 
f(x) dx = 0. 
Therefore, the probability that a crv will assume a fixed value is 0.
Thus, P(X < a) = P(X �
a), etc.
Note:� f(x) need not
be continuous although X is a crv.
Cumulative Distribution Function
The cumulative distribution function (cdf)
of a rv X is a function defined on R as follows:
Properties of F(x)

F(x) is defined for every real number x

0 � F(x) �
1

When x � �,
F(x) � 0.� When x �
�, F(x) �
1

F(x) is a nondecreasing function: if s < t, then
F(s) � F(t)

If X is a crv, then F(x) is a continuous function

P(a < X � b) =
F(b)  F(a)

If x is not an endpoint of an interval, then f(x) = F'(x)
Let m, l and u be the median, lower quartile and upper
quartile of X.� Then
F(m) = 0.5,� F(l) = 0.25,� F(u)
= 0.75.
Expectation
For a crv X with pdf f(x), the expectation
of X, written as E(X), is given by
E(X) is also denoted by m and referred
to as the mean of X.
Note:� If f(x) is symmetrical
about the central value c, then E(X) = c.
In general, if g(X) is any function of the random variable X,
then
E[g(X)] =� 
�
�_{all x}

g(t) f(t) dt. 

Properties of E �(similar to that for drv)
Let a and b be any constants

E(a) = a

E(aX) = aE(X)

E(aX + b) = aE(X) + b

E[f(X) � g(X)] = E[f(X)]
� E[g(X)]

E(XY) = E(X)E(Y)� if X and Y are
independent
Variance
The variance of a rv X, denoted by
Var(X), is defined as
Var(X) = E[(X  m)^{2}].
The standard deviation of X, denoted by s,
is the square root of Var(X):
Computational formula for Var(X):
Var(X) = E(X^{2}) 
[E(X)]^{2}.
Properties of Var �(similar to that for drv)

Var(a) = 0

Var(aX) = a^{2}Var(X)

Var(aX + b) = a^{2}Var(X)
Sum of Two Random Variables
If X and Y are any two random variables, then for any
constants a and b,�
E(aX � bY) = aE(X)
� bE(Y).�
If X and Y are also independent, then�
Var(aX � bY) =
a^{2}Var(X) + b^{2}Var(Y).
