1.1 Model probabilites

The short story is that the Beta distribution can be understood as representing a distribution of probabilities, that is, it represents all the possible values of a probability when we don’t know what that probability is.

1.2 Generalization of uniform

Give me a continuous and bounded random variable, em, except the Uniform distribution. That is another way to look at beta distribution, continuous and bounded between 0, 1; also the density is not flat. \[ X \sim Beta(a, b), \text{ where } a>0, \ b>0. \]

\[ f_X(x) = c \cdot x ^{a-1}(1-x)^{b-1}, \text{ where } x>0. \]

What is \(c\)? Just a normalization constant. We’ll get the value of \(c\) later.

2.1 Bank and Post Office Story

Let \(X\) be the waiting time at Bank,

\[ X \sim Gamma(n_1, \lambda) \]

Let \(Y\) be the waiting time at Post Office,

\[ Y \sim Gamma(n_2, \lambda) \]

Assume \(X\) and \(Y\) are independent.

Then, what is the distribution of the proportion \(\frac{X}{X+Y}\)?

Define \(T = X+Y\) be the total waiting time.

Clearly, \(T \sim Gamma(n_1+n_2, \lambda)\), proved by MGF.

Define \(W = \frac{X}{X+Y}\) , the proportion of waiting time at Bank to the total waiting time.

What is the distribution of \(W\)?

We need to derive \(f_W(w)\), but first let’s find the joint pdf \(f_{T,W}(t,w)\) \[ \begin{align} f_{T,W}(t,w) &= f_{X,Y}(x,y) \left | \frac{\partial(x,y)}{\partial(t,w)} \right|\\ &= \frac{1}{\Gamma(n_1)}\lambda^{n_1}x^{n_1 - 1}e^{-\lambda x} \frac{1}{\Gamma(n_2)}\lambda^{n_2}x^{n_2 - 1}e^{-\lambda y} \left|-t\right| \\ &= \lambda^{n_1+n_2}t^{n_1+n_2-1}e^{\lambda t} \frac{1}{\Gamma(n_1)\Gamma(n_2)}w^{n_1 - 1}(1-w)^{n_2-1} \\ &= \frac{\lambda^{n_1+n_2}t^{n_1+n_2-1}e^{\lambda t}}{\Gamma(n_1+n_2)} \frac{\Gamma(n_1+n_2)}{\Gamma(n_1)\Gamma(n_2)}w^{n_1 - 1}(1-w)^{n_2-1}\\ &= f_T(t) \frac{\Gamma(n_1+n_2)}{\Gamma(n_1)\Gamma(n_2)}w^{n_1 - 1}(1-w)^{n_2-1} \end{align} \]

Then we find the marginal,

Since \(f_W(w)\) is the pdf needed to be integrated to 1,

\[ \int_0^1\frac{\Gamma(n_1+n_2)}{\Gamma(n_1)\Gamma(n_2)}w^{n_1 - 1}(1-w)^{n_2-1} dw \equiv 1 \]

so the normalization constant should be

\[ c = \frac{\Gamma(n_1+n_2)}{\Gamma(n_1)\Gamma(n_2)} := \frac{1}{B(n_1, n_2)} \]

2.2 plots

library(zetaEDA)
library(ggfortify)
enable_zeta_ggplot_theme()

Let’s check Beta density for some different parameters value.

How about \(a = b = 1\)? The Beta(1,1) is just the Unif(0,1).

ggdistribution(func = dbeta, x = seq(0, 1, .01), shape1 = 1, shape2 = 1) +
  labs(title = "Beta Density with a = 1, b = 1")

How about \(a = b = \frac{1}{2}\)?

How about \(a = b = 2\)?

How about \(a= 2, \ b = 1\)?

One more,

p <- ggdistribution(func = dbeta, x = seq(0, 1, .01), shape1 = 1.5, shape2 = 5, colour = "tomato", linetype = "dashed")

ggdistribution(func = dbeta, x = seq(0, 1, .01), shape1 = 5, shape2 = 1.5, colour = "blue", p = p) +
  labs(title = "Red: a = 1.5, b = 5\n Blue: a = 5, b = 1.5")

For more checking, click this link and try some parameters to check the density curve.

Have fun!