Continuous Probability Distribution

Definition

Uniform Distribution

Questions:

If X1 and X2 both from Uniform Distribution, how about X1 + X2 , X1 * X2, max(X1, X2), min(X1, X2), 1-X1?

Normal Distribution

Normality Test includes Shapiro-Wilk W Test, Anderson-Darling Test(AD-Test), and Kolmogorov-Smirnov Test.

If log(x) is normally distributed, we say x has the lognormal distribution

Exponential Distribution

Model the time taken between the occurrence of different events.

Use cases include survival analysis(expected life of a device/machine), and specified number of defaults within a specified time period. In finance, it is often used to measure the likelihood of the next default for a portfolio of financial assets.

𝜆(lambda) is called the “rate” parameter, and is proportional to how quickly things happen. y is the PDF.

The exponential distribution has memoryless property. From the point of view of waiting time until arrival of a customer, the memoryless property means that it does not matter how long you have waited so far. Only exponential random variable has this property in continuous probability distribution.

Useful Link: https://www.youtube.com/watch?v=p3T-_LMrvBc

t-Distribution

t-distribution is useful for making inferences about population mean when 𝜎² 𝑖𝑠 unknown. When the degree of freedom is infinite, t-distribution = normal distribution.

Gamma Distribution

The family of exponential distributions is a subfamily of the gamma distributions. The times between successive occurrences in a Poisson process have an exponential distribution.

The gamma distribution comes up when modeling the time until the
next n events occur. It appears in machine learning as the “conjugate prior” .

T(α) is gamma function

For Gamma Function

Chi-square distribution is a special case of gamma distribution when α=m/2 and β=1/2, with m degree of freedom.

Beta Distribution

The beta distribution is best for representing a probabilistic distribution of probabilities - that is, it represents all the possible values of a probability when we don’t know what that probability is. This link gives a good explanation about the intuition behind Beta Distribution.

B is called Beta function

Beta Function

F Distribution

Arises frequently as the null distribution of a test statistic, most notably in F-tests associated with equality of variances and analysis of variance (ANOVA).

Weibull Distribution

Weibull distribution can model increasing (or decreasing) rates of failure over time, whereas the exponential distribution is appropriate when the rate — of wear, or failure for instance — is constant. Its application includes assessing product reliability, analyzing life data.

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