Continuous Random Variables and Probability Distributions

Probability Density Function:

Let X be a continuous random variable, and let x be any number lying in the range of values this random variable can take. The probability density function, f (x), of the random variable is a function with the following properties:

1.    f(x) > 0 for all values of x.
2.    The area under the probability density function, f(x), over all values of the random variable, X, is equal to 1.0.
3.    Suppose that this density function is graphed. Let a and b be two possible values of random variable X, with a < b. Then the probability that X lies between a and b is the area under the density function between these points.
4.    The cumulative distribution function, F(), is the area under the probability density function , f(x), up to  :

                         
where Xm  is the minimum value of the random variable x.

Areas Under Continuous Probability Density Functions

Let X be a continuous random variable with probability density function f(x) and cumulative distribution function F(x). Then the following properties hold:

1.    The total area under the curve f(x) is 1.
2.    The area under the curve f(x) to the left of  is F(), where  is any value that the random variable can take.

The Normal Distribution

The probability density function for a normally distributed random variable X is

The normal probability distribution represents a large family of distributions, each with a unique specification for the parameters  and . These parameters have a very convenient interpretation.

Properties of the Normal Distribution

Suppose that the random variable X follows a normal distribution with parameters   and  . Then the following properties hold:
1.    The mean of the random variable is

                    

2.    The variance of the random variable is :
                   Var(X) = 

3.    The shape of the probability density function is a symmetrical bell-shaped curve centered on the mean, , as shown in Figure 1.

4.    If we know the mean and variance, we can define the normal distribution by using the notation
                  

 

For our applied statistical analyses the normal distribution has a number of important characteristics. It is symmetric. Different central tendencies are indicated by differences in . In contrast, differences in  result in density functions of different widths. By selecting values for  and  we can define a large family of normal probability density functions. Differences in the mean result in shifts of entire distributions. In contrast, differences in the variance result in distributions with different widths.

Figure: Effects of   and   on the probability density function of a Normal random variable.

a.    Two normal distributions with different means.
b.    Two normal distributions with different variances and mean = 5.

Cumulative Distribution Function of the Normal Distribution

Suppose that X is a normal random variable with mean  and variance ; that is,  Then the cumulative distribution function is

                    
This is the area under the normal probability density function to the left of x0. As for any proper density function, the total area under the curve is 1; that is,

                                 
Figure: The shaded area is the probability that X does not exceed x0 for a normal random variable.

Range Probabilities for Normal Random Variables

Let X be a normal random variable with cumulative distribution function F(x), and let a  and b be two possible values of X, with a < b. Then

                        

The probability is the area under the corresponding probability density function between a and b, as shown in the following Figure.

Figure: Normal density function with the shaded area indicating the probability that X is between a and b.

The Standard Normal Distribution

Let Z be a normal random variable with mean 0 and variance 1; that is,
                              Z~N(0, 1)
We say that Z follows the standard normal distribution.

Denote the cumulative distribution function as F(z) and a and b as two numbers with a < b; then
                 P(a<Z<b)= F(b) – F(a)

We can obtain probabilities for any normally distributed random variable by first converting the random variable to the standard normally distributed random variable, Z. There is always a direct relationship between any normally distributed random variable and Z. That relationship buses the transformation

This important result allows us to use the standard normal table to compute probabilities associated with any normally distributed random variable. Now let us see how probabilities can be computed for the standard normal Z.

The cumulative distribution function of the standard normal distribution is tabulated in Table 1 in the Appendix. This table gives values of

                       F(z) = P(Z

for non-negative values of z. For example, the cumulative probability for a Z value of 1.25 from Table 1 is

                            F(1.25) = 0.8944

This is the area, designated in the following Figure, for Z less than 1.25. Because of the symmetry of the normal distribution, the probability that Z > -1.25 is also equal to .8944. In general, values of the cumulative distribution function for negative values of Z can be inferred using the symmetry of the probability density function.

To find the cumulative probability for a negative Z (for example, Z = -1.0) defined as

we use the complement of the probability for Z = +1, as shown in the following Figure.

From symmetry we can state that

The following Figure indicate the symmetry for the corresponding positive values of Z.

Figure: Standard normal distribution           Figure: Normal density function
for positive value Z equal 1                          with symmetric upper and lower values