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

 � �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:
For any real number x,
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)

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):

 ______ s = � Var(X) .

Computational formula for Var(X):

Var(X) = E(X2) - [E(X)]2.

Properties of Var �(similar to that for drv)

Let a and b be constants
• 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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