Continuous Random Variables

A continuous random variable is a random variable with a set of possible values that is infinite and uncountable, such as all real numbers. Choose one of the following major continuous distributions to explore.

The Bernoulli distribution is the probability distribution of a random variable which takes the value 1 with success probability of \(p\) and the value 0 with failure probability of \(1-p\). It is frequently used to represent binary experiments, such as a coin toss.

The binomial distribution describes the result of \(n\) independent Bernoulli trials with probability \(p\). It is frequently used to model the number of successes in a specific number of identical binary experiments, such as the number of heads in five coin tosses.

The negative binomial distribution is the probability distribution of the number of successes in a sequence of independent Bernoulli trials with probability \(p\) before \(r\) failures occur. For example this distribution could be used to model the number of heads before three tails in a sequence of coin tosses.

The geometric distribution is the probability distribution that the first success is after exactly \(k\) failures in a sequence of independent Bernoulli trials with probability \(p\). For example this distribution can be used to model the number of rolls of a dice before the first six is rolled.

The Poisson distribution describes the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate \(\lambda\), and independently of the time since the last event. This distribution has been used to model events such as meteor showers and goals in a soccer match.

The uniform distribution is a continuous distribution such that all intervals of the same length on the distribution's support are equally probable. For example this distribution might be used to model people's birth dates, where it is assumed all times are equally likely.

The normal (or Gaussian) distribution is used in the natural and social sciences to represent real-valued random variables whose distributions are not known. For example the normal distribution might be used to model people's height, where it is assumed most people are around the same height.

The Student's t-distribution, or simply the t-distribution, arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

The chi-squared distribution arises from the sum of \(k\) independent and identically distributed squared standard normal distributions. It is used frequently in hypothesis testing and in the construction of confidence intervals.

The exponential distribution is the continuous analogue of the geometric distribution. It is frequently used when describing the time between independent events such as modelling atomic decay.

The F-distribution, also known as the Fisher–Snedecor distribution arises frequently as the null distribution of a test statistic, most notably in the analysis of variance.

The gamma distribution is a general family of continuous probability distributions used frequently to model data that is skewed normal. The exponential distribution and chi-squared distribution are special cases of the gamma distribution.

The beta distribution is a general family of continuous probability distributions bound between 0 and 1. The beta distribution is frequently used as a conjugate prior distribution in Bayesian Statistics.

\(\large p\) = 0.5

\(\large n\) = 5
\(\large p\) = 0.5

\(\large r\) = 5
\(\large p\) = 0.5

\(\large p\) = 0.5

\(\large\lambda\) = 5

\(\large a\) = -5
\(\large b\) = 5

\(\large \mu\) = 0
\(\large \sigma\) = 1

\(\large k\) = 5

\(\large \lambda\) = 5

\(\large d_{1}\) = 5
\(\large d_{2}\) = 5

\(\large k\) = 5
\(\large \theta\) = 5

\(\large \alpha\) = 5
\(\large \beta\) = 5

PMF PDF	Mean	Variance
\(\left\{\begin{array}{ll} (1-p) \text{ for } x = 0\\ p \qquad \text{ for } x = 1 \end{array}\right.\) \(\binom{n}{x}p^{x}(1-p)^{n-x}\) \(\binom{x + r -1}{x}p^{x}(1-p)^{r}\) \((1-p)^{k}p\) \(\dfrac{\lambda^{x}e^{-\lambda}}{x!}\) \(\left\{\begin{array}{ll} \dfrac{1}{b-a} \text{ for } x \in [a,b]\\ 0 \qquad \text{ otherwise } \end{array}\right.\) \(\dfrac{1}{\sqrt{2\sigma^{2}\pi}} e^{-\dfrac{(x-\mu)^{2}}{2\sigma^{2}}}\) \(\dfrac{Z}{\sqrt{U/k}} \qquad \begin{array}{ll} Z \sim N(0,1)\\ U \sim \chi_{k} \end{array}\) \(\sum_{i=1}^{k}Z_{i}^{2} \qquad Z_{i} \overset{i.i.d.}{\sim} N(0,1)\) \(\lambda e^{-\lambda x}\) \(\dfrac{U_{1}/d_{1}}{U_{2}/d_{2}} \qquad \begin{array}{ll} U_{1} \sim \chi_{d_{1}}\\ U_{2} \sim \chi_{d_{2}} \end{array}\) \(\dfrac{1}{\Gamma(k)\theta^{k}}x^{k-1}e^{-\dfrac{x}{\theta}}\) \(\dfrac{\Gamma(\alpha + \beta)x^{\alpha - 1}(1-x)^{\beta - 1}}{\Gamma(\alpha)\Gamma(\beta)}\)	\(p\) \(np\) \(\dfrac{pr}{1-p}\) \(\dfrac{1-p}{p}\) \(\lambda\) \(\dfrac{a+b}{2}\) \(\mu\) \(0\) \(k\) \(\lambda^{-1}\) \(\dfrac{d_{2}}{d_{2}-2}\) \(k\theta\) \(\dfrac{\alpha}{\alpha + \beta}\)	\(p(1-p)\) \(np(1-p)\) \(\dfrac{pr}{(1-p)^{2}}\) \(\dfrac{1-p}{p^{2}}\) \(\lambda\) \(\dfrac{(b-a)^{2}}{12}\) \(\sigma^{2}\) \(\dfrac{k}{k-2}\) \(2k\) \(\lambda^{-2}\) \(\dfrac{2d_{2}^{2}(d_{1} + d_{2} -2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}\) \(k\theta^{2}\) \(\dfrac{\alpha\beta}{(\alpha + \beta)^{2}(\alpha + \beta + 1)}\)