Statistical Models: Beta Distribution

4 Common Distributions

Binomial Distribution

$P(x = k) = C_n^k p^k (1 - p)^{n - k}$ : In $n$ independent trials, with success probability $p$ each time, until $k$ successes occur.

• Expectation $E = np$

• Variance $Var = np(1 - p)$

• Cumulative probability density function $F(x \le k) = \sum_{x = 0}^k C_n^k p^k (1 - p)^{n - k}$

Application scenarios:

1. Quality control

2. Reliability

3. Survey sampling

In certain cases, it can be approximated by other distributions:

1. As $p \to 0$, $n \to \infty$, it can be approximated by the Poisson distribution.

2. If $p \le 0.5$, $np > 5$, or $p > 0.5$, $nq > 5$, it can be approximated by the normal distribution.

Poisson Distribution

$P(x = k) = \frac{\exp( - \lambda) \lambda^k }{k!}$, denoted as $x \sim \pi(\lambda)$, where $\lambda$ is the average number of event occurrences.

Describes the probability distribution of the number of times a random event (with constant probability) occurs per unit time.

• Expectation $E = \lambda$

• Variance $Var = \lambda$

• $X + Y \sim \pi(\lambda_1 + \lambda_2)$

Application scenarios:

1. For example, the number of service requests received by a service facility within a certain period.

2. The number of calls received by a telephone exchange.

3. The number of passengers waiting at a bus stop.

4. The number of machine failures.

Exponential Distribution

$p(T = t) = \lambda \exp(- \lambda t)$

Where $\lambda > 0$ is the parameter of the distribution, i.e., the number of times the event occurs per unit time.

Used to analyze the probability distribution of intervals between independent events, such as the time interval between two tourists entering an airport (if the interval is likely to exceed one hour, airport staff can happily slack off).

• Expectation $E = \frac{1}{\lambda}$

• Variance $Var = \frac{1}{\lambda^2}$

• Memoryless property (an excellent property): $P(T > s + t | T > t) = P(T > s)$

Application scenarios:

• Estimating the lifespan of light bulbs.

Normal Distribution

$P(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp(\frac{- (x - \mu)^ 2}{ 2 \sigma ^ 2})$

Beta Distribution

$p(x | \lambda_1, \lambda_2, w_1, w_2) = \frac{1}{w_2 - w_1} \frac{\Gamma(\lambda_1 + \lambda_2)}{\Gamma(\lambda_1) \lambda_2} (\frac{x - w_1}{w_2 - w_1})^{\lambda_1 - 1} (1 - \frac{x - w_1}{w_2 - w_1})^{\lambda_2 - 1}$

The Beta distribution is used to describe the probability distribution of probabilities (quite a mouthful). Since probabilities lie in $[0, 1]$, the range of the Beta distribution is also $[0, 1]$, and it is used to model probabilities.

Typically, $w_1 = 0, w_2 = 1$, so $p(x | \lambda_1, \lambda_2) = \frac{\Gamma(\lambda_1 + \lambda_2)}{\Gamma(\lambda_1) \Gamma(\lambda_2)} x^{\lambda_1 - 1} (1 - x)^{\lambda_2 - 1}$

The Beta function $B(\alpha, \beta) = \int_0^1 x^{a - 1} (1 - x )dx = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)}$

Beta distribution:

$$f(x) = \frac{1}{\Beta(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, 0< x < 1$$

The Beta distribution and the Binomial distribution are very similar in form in terms of their probability density functions. The difference is that the Binomial distribution models the number of successful trials, while the Beta distribution models the success probability itself, analyzing the probability of a success probability (still quite a mouthful).

• Expectation $E = \frac{\alpha }{\alpha + \beta}$

• Variance $Var = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$

Application scenarios:

• Bayesian analysis (used to describe the prior distribution).