# 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.

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” .

## 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.

## 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.

**Thanks for reading the post. Hope it is useful!**