Notes - Probability MT22, Distributions
Discrete
Uniform
If $X$ is discrete and uniformly distributed, i.e. $X \sim U\{1, 2, \ldots, n\}$, what is the probability mass function $\mathbb{P}(X = k)$?
If $X$ is discrete and uniformly distributed, i.e. $X \sim U\{1, 2, \ldots, n\}$, what is the mean $\mathbb{E}[X]$?
If $X$ is discrete and uniformly distributed, i.e. $X \sim U\{1, 2, \ldots, n\}$, what is the variance $\text{var}(X)$?
If $X$ is discrete and uniformly distributed, i.e. $X \sim U\{1, 2, \ldots, n\}$, what is the generating function $G _ X(s)$?
Bernoulli
If $X$ is discrete and Bernoulli distributed, i.e. $X \sim \text{Ber}(p)$, what is the probability mass function $\mathbb{P}(X = k)$?
If $X$ is discrete and Bernoulli distributed, i.e. $X \sim \text{Ber}(p)$, what is the mean $\mathbb{E}[X]$?
If $X$ is discrete and Bernoulli distributed, i.e. $X \sim \text{Ber}(p)$, what is the variance $\text{var}(X)$?
If $X$ is discrete and Bernoulli distributed, i.e. $X \sim \text{Ber}(p)$, what is the generating function $G _ X(s)$?
Binomial
If $X$ is discrete and binomially distributed, i.e. $X \sim \text{Bin}(n, p)$, what is the probability mass function $\mathbb{P}(X = k)$?
If $X$ is discrete and binomially distributed, i.e. $X \sim \text{Bin}(n, p)$, what is the mean $\mathbb{E}[X]$?
If $X$ is discrete and binomially distributed, i.e. $X \sim \text{Bin}(n, p)$, what is the variance $\text{var}(X)$?
If $X$ is discrete and binomially distributed, i.e. $X \sim \text{Bin}(n, p)$, what is the generating function $G _ X(s)$?
Poisson
If $X$ is discrete and Poisson distributed, i.e. $X \sim \text{Poi}(\lambda)$, what is the probability mass function $\mathbb{P}(X = k)$?
If $X$ is discrete and Poisson distributed, i.e. $X \sim \text{Poi}(\lambda)$, what is the mean $\mathbb{E}[X]$?
If $X$ is discrete and Poisson distributed, i.e. $X \sim \text{Poi}(\lambda)$, what is the variance $\text{var}(X)$?
What is special about the Poisson distribution’s $\text{Poi}(\lambda)$ mean and variance?
It is the same, $\mathbb{E}[X] = \text{var}(X) = \lambda$.
If $X$ is discrete and Poisson distributed, i.e. $X \sim \text{Poi}(\lambda)$, what is the generating function $G _ X(s)$?
Geometric ($k$ total)
If $X$ is discrete and geometrically distributed (in the case of $k$ total trials for the first success), i.e. $X \sim \text{Geom}(p)$, what is the probability mass function $\mathbb{P}(X = k)$?
If $X$ is discrete and geometrically distributed (in the case of $k$ total trials for the first success), i.e. $X \sim \text{Geom}(p)$, what is the mean $\mathbb{E}[X]$?
If $X$ is discrete and geometrically distributed (in the case of $k$ total trials for the first success), i.e. $X \sim \text{Geom}(p)$, what is the variance $\text{var}(X)$?
If $X$ is discrete and geometrically distributed (in the case of $k$ total trials for the first success), i.e. $X \sim \text{Geom}(p)$, what is the generating function $G _ X(s)$?
Alternative Geometric ($k$ failures)
If $X$ is discrete and geometrically distributed (in the case of $k$ failures for the first success), i.e. $X \sim \text{Geom}(p)$, what is the probability mass function $\mathbb{P}(X = k)$?
If $X$ is discrete and geometrically distributed (in the case of $k$ failures for the first success), i.e. $X \sim \text{Geom}(p)$, what is the mean $\mathbb{E}[X]$?
If $X$ is discrete and geometrically distributed (in the case of $k$ failures for the first success), i.e. $X \sim \text{Geom}(p)$, what is the variance $\text{var}(X)$?
If $X$ is discrete and Poisson distributed (in the case of $k$ failures for the first success), i.e. $X \sim \text{Geom}(p)$, what is the generating function $G _ X(s)$?
Negative Binomial
If $X$ is discrete and negatively binomially distributed, i.e. $X \sim \text{NegBin}(k, p)$, what is the interpretation of $\mathbb{P}(X = n)$?
The probability that there are $n$ failures before there a total of $k$ successes.
Why are the expressions for the expectation and variance of $X \sim \text{NegBin}(k, p)$ equal to $k$ times the expectation and variance of a geometrically distributed random variable?
Because the negatively binomial distribution can be considered a sum of $k$ geometric random variables.
What’s the similarity between the binomial distribution and the negative binomial distribution?
- Binomial: Models value of sum of Bernoulli variables
- Negative binomial: Models the number of total failures before a certain number of successful Bernoulli trials