Discrete Random Variables

26 Jun 2001

Main Points

Definition
Probability Density Function
Cumulative Distribution Function
Expectation
Variance
Sum of Two Random Variables

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

�
�_{all x}

f(x) dx = 1,

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:

F(x) =�

P(X � x)

=�

�^x
�_-�

f(t) dt.

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 non-decreasing 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) =�

� �_{all x}

t f(t) dt.

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)

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):

	______
s = �	Var(X) .

Computational formula for Var(X):

Var(X) = E(X²) - [E(X)]².

Properties of Var �(similar to that for drv)

Var(a) = 0
Var(aX) = a²Var(X)
Var(aX + b) = a²Var(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) = a²Var(X) + b²Var(Y).

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