4.Continuous Random Variables and Probability Distribution

1. Continuous Random Variables and Probability Density Functions

A random variable whose set of possible values is an entire interval of numbers is not discrete.

Continuous Random Variables

A random variable X is said to be continuous if its set of possible values is an entire interval of numbers -- that is, if for some A<B, any number x between A and B is possible.

Probability Distribution of Continuous Variables

Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function ƒ(x) such that for any two numbers a and b with a≤b,

P(a≤X≤b) = ∫ƒ(x)dx

That is, the probability that X takes on a value in the interval [a,b] is the area under the graph of the density function. The graph of ƒ(x) is often referred to as the density curve.

A continuous rv X is said to be uniform distribution on the interval [A,B] if the pdf of X is 

ƒ(x;A,B) = 1/(B-A) for A≤x≤B

             = 0 for otherwise

 

2. Cumulative Distribution Functions and Expected Values

The Cumulative Distribution Function

The cumulative distribution function F(x) for a continuous rv X defined for every number x by

F(x) = p(X≤x) = ∫ƒ(x)dx

For each x, F(x) is the area under the density curve to the left of x, where F(x) increases smoothly as x increases.

Using F(x) to Compute Probabilities

The importance of the cdf here, just as for discrete rv's, is that probabilities of various intervals can be computed from a formula for or table of F(x).

PROPOSITION:

Let X be a continuous rv with pdf ƒ(x) and cdf F(x).

Then for any number a,

　　P(X>a) = 1 - F(a)

and for any two numbers a and b with a<b,

　　P(a≤X≤b) = F(b) - F(a)

Obtaining ƒ(x) from F(x)

For  X discrete, the pmf is obtained from the cdf by taking the difference between two F(x) values. The continuous analog of a difference is a derivative.

PROPOSITION:

If X is a continuous rv with pdf ƒ(x) and cdf F(x), then at every x at which the derivative F'(x) exists, F'(x) = ƒ(x).

Percentiles of a Continuous Distribution

Let p be a number between 0 and 1. The (100p)th percentile of the distribution of a continuous rv X, denoted by η(p), is defined by p = F(η(p))

Expected Values for Continuous Random Variables

The expected or mean value of a continuous rv X with pdf ƒ(x) is μ_x = E(X) = ∫xƒ(x)dx

If X is a continuous rv with pdf ƒ(x) and h(X) is any function of X, then E[h(x)] = μ_h(x) = ∫h(x)ƒ(x)dx

The Variance of a Continuous Random Variable

V(X) = E(X²) - [E(X)]²

 

3. The Normal Distribution

A continuous rv X is said to have a normal distribution with parameters μ and σ, where -∞<μ<∞ and 0<σ

The Standard Normal Distribution

The normal distribution with parameter values μ=0 and σ=1 is called a standard normal distribution. A random variable that has a standard is called a standard normal random variable and will be denoted by Z.

Percentiles of the Standard Normal Distribution

For any p between 0 and 1, Appendix Table can be used to obtain the 100pth percentile of the standard normal distribution.

Z_α Notation

Z_α will denote the value on the measurement axis for which α of the area under the z curve lies to the right of z_α

Nonstandard Normal Distributions

If X has a normal distribution with mean μ and standard deviation σ, then Z = (X-μ)/σ

If the population distribution of a variable is (approximately) normal, then 

1. Roughly 68% of the values are within 1 SD of the mean.
2. Roughly 95% of the values are within 2 SDs of the mean.
3. Roughly 99.7% of the values are within 3 SDs of the mean.

Percentiles of an Arbitrary Normal Distribution

(100p)th percentile for normal (μ,σ) = μ + [(100p)th for standard normal]*σ

The Normal Distribution and Discrete Population

The normal distribution is often used as an approximation to the distribution of values in a discrete population. The correction for discreteness of the underlying distribution is called a continuity correction. It is helpful in the following application of the normal distribution to the computation of binomial probabilities.

The Normal Approximation to the Binomial Distribution

As long as the binomial probability histogram is not too skewed, binomial probabilitites can be well approximated by normal curve areas. It is then customary to say that X has apporoximately a normal distribution.

 

4. The Gamma Distribution and Its Relatives

For α>0, the gamma function Γ(α) is defined by Γ(α) = 

5. Other Continuous Distribution