26 Jun 2001

Main Points


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
f(x) dx.

If a = b, then

P(X = a) = P(a X 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)
f(t) dt.

Properties of 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.


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)


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)

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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