Study for Statistics Test Level 2 (7)
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Today's study is about specific distributions.
Discrete probability distributions
Bernoulli distribution
A trial is termed a "Bernoulli trial" when the outcomes are independent of each other and have the same probability.
When only one Bernoulli trial is conducted, and there are two possible outcomes, the resulting distribution is called a "Bernoulli distribution." Its probability function is defined as:
\[ P(X = k) = p^k (1 - p)^{1 - k} \]
Binomial distribution
A "Binomial distribution" arises when one outcome occurs times and the other outcome occurs times in Bernoulli trials.
The probability function for a binomial distribution is defined as:
\[ P(X = k) = \frac{n!}{k! (n - k)!}p^k (1 - p)^{n - k} \]
expectation：\[ \mu = np \]
variation：\[ σ^2 = np(1 - p) \]
standard deviation：\[ σ = \sqrt{np(1 - p)} \]
Poisson distribution
In a binomial distribution, when the expectation is defined as a specific value, and approaches infinity while approaches zero, the resulting distribution is termed a "Poisson distribution."
The Poisson distribution serves as a useful probability model for events that occur rarely.
probability function：\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]
expectation：\[ \mu = \lambda \]
variation：\[ σ^2 = \lambda \]
Geometric distribution
A geometric distribution describes the number of Bernoulli trials needed for the first success.
probability function：\[ P(X = k) = (1 - p)^{k - 1} p \]
expectation：\[ \mu = \frac{1}{p} \]
variation：\[ σ^2 = \frac{1 - p}{p^2} \]
Continuous probability distributions
Uniform distribution
A uniform distribution has equal probability across the range of to .
probability density function：\[ f(x) = \begin{cases} \frac{1}{b - a} & \text{for } a \leq x \leq b, \\ 0 & \text{otherwise.} \end{cases} \]
expectation：\[ \mu = \frac{a + b}{2} \]
variation：\[ σ^2 = \frac{(b - a)^2}{12} \]
Normal distribution(Gaussian distribution)
When the expectation and variance are defined, the probability function of a normal distribution is given as:
probability density function：\[ f(x) = \frac{1}{σ\sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x - \mu}{σ}\right)^2} \]
standardization：\[ z = \frac{x - \mu}{σ} \]
If random variables follow a normal distribution, their sum also follows a normal distribution.
Exponential distribution
The exponential distribution represents the time until the first event occurs in a Bernoulli trial.
probability density function：\[ f(x) = \lambda e^{-\lambda x} \text{ for } x \geq 0, \text{ and } 0 \text{ otherwise.} \]
expectation：\[ \mu = \frac{1}{\lambda} \]
variation：\[ σ^2 = \frac{1}{\lambda^2} \]
Today's study has concluded.
