Mar 17

Examples of distribution functions

Examples of distribution functions: the first two are used in binary choice models, the third one is applied in maximum likelihood.

Example 1. Distribution function of a normal variable

The standard normal distribution z is defined by its probability density


It is nonnegative and integrates to 1 (the proof of this fact is not elementary). Going from density function to distribution function gives us the distribution function (cdf) of the standard normal:

F_z(x)=\int_{-\infty}^xp_z(t)dt, for all real x.

Example 2. The logistic distribution

Here we go from distribution function to density function.

Consider the function


It's easy to check that it has the three characteristic properties of a distribution function: the limits at the right and left infinities and monotonicity.

1. When x\rightarrow\infty1+e^{-x} goes to 1, so F(x) tends to 1.

2. If x\rightarrow -\infty1+e^{-x} goes to +\infty, and F(x) tends to 0.

3. Finally, to check monotonicity, we can use the following sufficient condition: a function is increasing where its derivative is positive. (From the Newton-Leibniz formula f(x_2)=f(x_1)+\int_{x_1}^{x_2}f'(t)dt we see that positivity of the derivative and {x_2}>x_1 imply f(x_2)>f(x_1)). The derivative

(1) F'(x)=\frac{e^{-x}}{(1+e^{-x})^2}

is positive, so F(x) is increasing.

Thus, F is a distribution function, and it generates a density (1).

Example 3. Distribution function and density of a discrete variable

The distribution function concept applies to all random variables, both discrete and continuous. For discrete variables, the distribution function is not continuous as in Figure 1 here; it has jumps at points that have a positive probability attached. We illustrate this using a Bernoulli variable B such that P(B=0)=0.4 and P(B=1)=0.6.

  1. For x<0 we have F_B(x)=P(B\le x)=0.
  2. For 0\le x<1 we have F_B(x)=P(B=0)+P(0<B\le x)=0.4.
  3. Finally, F_B(x)=P(B=0)+P(B=1)+P(1<B\le x)=1 for 1\le x<\infty.

This leads us to Figure 1.

Figure 1. Distribution function of the Bernoulli variable

Now consider B such that P(B=1)=p and P(B=0)=1-p. The analog of the density function for Bernoulli looks like this:

(2) p(x)=p^x(1-p)^{1-x}, for x=0,1.

To understand this equation, check that p(1)=p and p(0)=1-p. In Math, there are many tricks like this.

Remark. For continuous random variables, the value of the density at a fixed point means nothing (in particular, it can be larger than 1). It is its integral that has probabilistic meaning. For (2) the value of the density at a fixed point IS probability.

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