Probability Laws

The notion of the probability law of a random phenomenon is introduced in this section in order to provide a concise and intuitively meaningful language for describing the probability properties of a random phenomenon.

In order to describe a numerical valued random phenomenon, it is necessary and sufficient to state its probability function ; this is equivalent to stating for any Borel set of real numbers the probability that an observed value of the random phenomenon will be in the Borel set . However, other functions exist, a knowledge of which is equivalent to a knowledge of the probability function. The distribution function is one such function, for between probability functions and distribution functions there is a one-to-one correspondence. Similarly, between discrete distribution functions and probability mass functions and between continuous distribution functions and probability density functions one to-one correspondences exist. Thus we have available different, but equivalent, representations of the same mathematical concept, which we may call the probability law (or sometimes the probability distribution) of the numerical valued random phenomenon .

A probability law is called discrete if it corresponds to a discrete distribution function and continuous if it corresponds to a continuous distribution function.

For example, suppose one is considering the numerical valued random phenomenon that consists in observing the number of hits in five independent tosses of a dart at a target, where the probability at each toss of hitting the target is some constant . To describe the phenomenon, one needs to know, by definition, the probability function , which for any set of real numbers is given by

It should be recalled that represents the intersection of the sets and .

Equivalently, one may describe the phenomenon by stating its distribution function ; this is done by giving the value of at any real number , It should be recalled that denotes the largest integer less than or equal to .

Equivalently, since the distribution function is discrete, one may describe the phenomenon by stating its probability mass function , given by \begin{align} p(x) = \begin{cases} \displaystyle \binom{5}{x} p^{x} q^{5-x}, & \text{for } x = 0, 1, \ldots, 5 \ Examples of random phenomena obeying a uniform probability-law are discussed in section 5.

The normal probability law with parameters and , where and , is specified by the probability density function  

The role played by the normal probability law in probability theory is discussed in Chapter 6 . In section 6 we introduce certain functions that are helpful in the study of the normal probability law.

The exponential probability law with parameter , in which , is specified by the probability density function \begin{align} f(x) = \begin{cases} \lambda e^{-\lambda x}, & \text{for } x > 0 \ 

Student’s distribution with parameter (also called Student’s -distribution with degrees of freedom) is specified by the probability density function  

It should be noted that Student’s distribution with parameter coincides with the Cauchy probability law with parameters and .

The distribution with parameters and is specified by the probability density function \begin{align} f(x) = \begin{cases} \frac{1}{2^{n/2} \sigma^{n} \Gamma(n/2)} x^{(n/2)-1} e^{-\left(x / 2 \sigma^{2}\right)}, & \text{for } x > 0 \\[2mm] 0, & \text{for } x < 0 \end{cases} \tag{4.16} \end{align} 

The symbol is the Greek letter chi, and one sometimes writes chi-square for . The distribution with parameters and is called in statistics the distribution with degrees of freedom. The distribution with parameters and coincides with the gamma distribution with parameters and [to define the gamma probability law for non-integer , replace ! in (4.13) by .

The distribution with parameters and is specified by the probability density function \begin{align} f(x) = \begin{cases} \frac{2(n/2)^{n/2}}{\sigma^{n} \Gamma(n/2)} x^{n-1} e^{-\left(\frac{n}{2} \sigma^{2}\right) x^{2}}, & \text{for } x > 0 \\[2mm] 0, & \text{for } x < 0. \end{cases} \tag{4.17} \end{align} 

The distribution with parameters and is often called the chi distribution with degrees of freedom. (The relation between the and distributions is given in exercise 8.1 of Chapter 7 ).

The Rayleigh distribution with parameter is specified by the probability density function \begin{align} f(x) = \begin{cases} \frac{1}{\alpha^{2}} x e^{-\frac{1}{2} \left(\frac{x}{\alpha}\right)^{2}}, & \text{for } x > 0 \\[2mm] 0, & \text{for } x < 0. \end{cases} \tag{4.18} \end{align} The Rayleigh distribution coincides with the distribution with parameters and .

The Maxwell distribution with parameter is specified by the probability density function \begin{align} f(x) = \begin{cases} \frac{4}{\sqrt{\pi}} \frac{1}{\alpha^{3}} x^{2} e^{-\frac{x^{2}}{\alpha^{2}}}, & \text{for } x > 0 \\[2mm] 0, & \text{for } x < 0. \end{cases} \tag{4.4.19} \end{align} 

The Maxwell distribution with parameter coincides with the distribution with parameter and .

The distribution with parameters and is specified by the probability density function \begin{align} f(x) &= \begin{cases} \displaystyle \frac{\Gamma\left(\dfrac{m+n}{2}\right)}{\Gamma\left(\dfrac{m}{2}\right) \Gamma\left(\dfrac{n}{2}\right)} \left(\dfrac{m}{n}\right)^{m / 2} \frac{x^{(m / 2)-1}}{\left[1+\left(\dfrac{m}{n}\right) x\right]^{(m+n) / 2}} & \text{for } x > 0, \tag{4.20} \\ 0 & \text{for } x < 0. \end{cases} \end{align} 

The beta probability law with parameters and , in which and are positive real numbers, is specified by the probability density function \begin{align} f(x) = \begin{cases} \displaystyle \frac{1}{B(a, b)} x^{a-1}(1-x)^{b-1}, & \text{for } 0 < x < 1 \\[2mm] 0, & \text{elsewhere.} \end{cases} \tag{4.21} \end{align} 

Theoretical Exercises

Exercises