## Chapter 4    Continuous Distributions

### 4.1 Continuous Random Variables and Densities

Recall: X is a continuous random variable if the possible values for X form a continuous range or interval.

ex:   Let L = length of leaf picked from tree; could take on any value in interval [0, 10]

Important note:

• It doesn’t make much sense to ask “what’s probability  L = 2.5 inches” .  The chance that L = exactly 2.500000000000...   inches should be  0.
• For a continuous random variable, the probability that X takes on any one specific value is zero:

•          P(X=x) = 0    for any x.
Instead, we'll ask for the probability that the value of X will lie in some range of values, usually an interval:  want to know the probability that the value of X lies between two specified values a and b,  P(a <= X <= b).

Can characterise continuous random variables with density function f(x); however, it will have a very different meaning from its interpretation in the discrete case!

Def:
Let X be a continous random variable. Then the density function  f  of  X  is the function for which

i.e., the probability that X lies between a & b is the area under the graph of f (x) from a to b.

The density function for a continuous random variable must satisfy 2 properties:

1. f(x) >= 0 for all x

2.   this says the height of the graph must be >= 0 for all x

3.   this says the probability that X lies between - and must equal 1:  X has to take on some value!

Note:  The probability of X taking on one specific value should be 0.  Indeed, by the above definition of the density function,

P(X=a)  =  P(a <= X <= a)  =    =   0.

ex:

Let X = weight of cereal in a randomly selected box; suppose density function of X is

The graph of f is:

What’s probability X lies between 14.5 and 15.5 ounces?
P(14.5 <= X <= 15)   =      =
=   0  -  (- (.5)2)   =   .25
What’s probability X <= 14.2?
P(X <= 14.2)   =   P(- < X <= 14.2)   =

(since f(x) = 0  for  x <= 14)
=   .36
What’s probability X >= 16?
P(X >= 16)   =   P(16 <= X < )   =    =   0;   since f(x) = 0 for x > 15, the area under the curve from  x = 16  to  x =    is 0.

Can characterize a continuous random variable by its cumulative distribution fn:

Def: The cumulative distribution function  F of a continuous random variable X is defined to be

F(x) = P(X <= x)
• notice that this is the same definition used in the discrete case.
• also called the cumulative probability function
If the density function of X is f(x), then
,
since P(X <= x)  equals the area under the graph of  f  to the left of x.

ex:

For the example involving the weight of cereal boxes above, the cumulative distribution function is found as follows:
for x <= 14,   F(x) = 0;   this follows because  P(X <= x) = 0 for any x less than 14, since the density function f(x) = 0 to the left of 14 and hence the area under the curve to the left of x is 0.

for x >= 15,   F(x) = 1;   this follows because  P(X <= x) = 1  for any x greater than 15, since the entire "hump" of the density function will lie to the left of x and hence the area under the curve to the left of x will be 1.

for x between 14 and 16, we have

F(x)   =   P(- <= X <= x)   =
(since f(x) = 0 for x less than 14)
=   1  -  (15 - x)2
Thus we have

The graph of  F  is:

The distribution function in the example above illustrates certain features shared by all cumulative distribution functions:

Properties of cumulative distribution functions

1.   0 <= F(x) <= 1   for all x
2.
3.
Note:  F(x) will be continuous!

Previous section  Next section