Distribution functions and density functions
Distribution functions
- a random variable
on the probability space is a mapping such that we abbreviate events of the form to the simpler expression .the function is said to be -measurable. - Whereas discrete random variables were studied via their mass functions, random variables in the broader sense are studied via their distribution functions, defined as follows.
- Definition 5.2 If
is a random variable on , the distribution function of is the function defined by - properties:
,which is to say that is monotonic non-decreasing , is continuous from the right, which is to say that - For
,
Examples of distribution functions
- Example 5.15 (Uniform distribution)Let
and . The function - Example 5.16 (Exponential distribution)Let
and let be given by
Continuous random variables
- Random variables come in many shapes, but there are two classes of
random variables which are particularly important:
- discrete random variables
- continuous random variables
- Discrete random variables take only countably many values, and their distribution functions generally look like step functions At the other extreme, there are random variables whose distribution functions are very smooth , and we call such random variables ‘continuous’.
- Definition 5.20:A random variable
is continuous if its distribution function may be written in the form for some non-negative function .In this case, we say that has (probability) density function (or pdf) . - If
is small and positive, then, roughly speaking, the probability that is ‘near’ satisfies So the true analogy is not between a density function and a mass function but instead between and . - Theorem 5.27 If
is continuous with density function , then and - To recap, all random variables have a distribution function. In addition, discrete random variables have a mass function, and continuous random variables have a density function.There are many random variables which are neither discrete nor continuous, and we shall come across some of these later.
Some common density functions
The uniform distribution on the interval
has density function The exponential distribution with parameter λ > 0 has density function
The normal (or Gaussian) distribution with parameters
and , sometimes written as , has density function The Cauchy distribution has density function
The gamma distribution with parameters
and has density function where is the gamma function. The beta distribution with parameters
has density function The beta function: is chosen so that has integral equal to one. You may care to prove that If , then is uniform on . The chi-squared distribution with n degrees of freedom (sometimes written
) has density function # Functions of random variables Theorem 5.50 If
is a continuous random variable with density function , and is a strictly increasing and differentiable function from into , then has density function where is the inverse function of . If, g were strictly decreasing
Expectations of continuous random variables
- Definition 5.56 If
is a continuous random variable with density function , the expectation of is denoted by and defined by whenever this integral converges absolutely, in that - As in the case of discrete variables, the expectation of X is often
called the expected value or mean of
. - Theorem 5.58 (Law of the subconscious statistician)
If
is a continuous random variable with density function , and , then whenever this integral converges absolutely.
Geometrical probability
The paradox of Joseph Louis Francois Bertrand
A chord of the unit circle is picked at random. What is the probability that an equilateral triangle with the chord as base fits inside the circle?
The needle of Georges Louis Leclerc, Comte de Buffon. This is more interesting.
A plane is ruled by straight lines which are unit distance apart, as in Figure 5.5. What is the probability that a unit needle, dropped at random, intersects a line?
Stick breaking.
Here is an everyday problem of broken sticks. A stick of unit length
is broken at the two places
