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 $\mathbb{P}(X = k)$?


\[\mathbb{P}(X = k) = \frac{1}{n}\]

If $X$ is discrete and uniformly distributed, i.e. $X \sim U\{1, 2, \ldots, n\}$, what is the mean $\mathbb{E}[X]$?


\[\frac{n+1}{2}\]

If $X$ is discrete and uniformly distributed, i.e. $X \sim U\{1, 2, \ldots, n\}$, what is the variance $\text{var}(X)$?


\[\frac{n^2 - 1}{12}\]

If $X$ is discrete and uniformly distributed, i.e. $X \sim U\{1, 2, \ldots, n\}$, what is the generating function $G _ X(s)$?


\[\frac{s - s^{n+1 } } {n(1-s)}\]

Bernoulli

If $X$ is discrete and Bernoulli distributed, i.e. $X \sim \text{Ber}(p)$, what is the probability mass function $\mathbb{P}(X = k)$?


\[\mathbb{P}(X = k) = \begin{cases} p &\text{if } k =1\\\\{1-p} &\text{if } k=0\end{cases}\]

If $X$ is discrete and Bernoulli distributed, i.e. $X \sim \text{Ber}(p)$, what is the mean $\mathbb{E}[X]$?


\[p\]

If $X$ is discrete and Bernoulli distributed, i.e. $X \sim \text{Ber}(p)$, what is the variance $\text{var}(X)$?


\[p(1-p)\]

If $X$ is discrete and Bernoulli distributed, i.e. $X \sim \text{Ber}(p)$, what is the generating function $G _ X(s)$?


\[1 - p + ps\]

Binomial

If $X$ is discrete and binomially distributed, i.e. $X \sim \text{Bin}(n, p)$, what is the probability mass function $\mathbb{P}(X = k)$?


\[\mathbb{P}(X = k) = \left(\begin{matrix}n \\\\k\end{matrix}\right)p^k(1-p)^{n-k}\]

If $X$ is discrete and binomially distributed, i.e. $X \sim \text{Bin}(n, p)$, what is the mean $\mathbb{E}[X]$?


\[np\]

If $X$ is discrete and binomially distributed, i.e. $X \sim \text{Bin}(n, p)$, what is the variance $\text{var}(X)$?


\[np(1-p)\]

If $X$ is discrete and binomially distributed, i.e. $X \sim \text{Bin}(n, p)$, what is the generating function $G _ X(s)$?


\[(ps + (1-p))^n\]

Poisson

If $X$ is discrete and Poisson distributed, i.e. $X \sim \text{Poi}(\lambda)$, what is the probability mass function $\mathbb{P}(X = k)$?


\[\mathbb{P}(X = k) = \frac{\lambda^k}{k!} e^{-\lambda}\]

If $X$ is discrete and Poisson distributed, i.e. $X \sim \text{Poi}(\lambda)$, what is the mean $\mathbb{E}[X]$?


\[\lambda\]

If $X$ is discrete and Poisson distributed, i.e. $X \sim \text{Poi}(\lambda)$, what is the variance $\text{var}(X)$?


\[\lambda\]

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 $G _ X(s)$?


\[e^{\lambda(s-1)}\]

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 $\mathbb{P}(X = k)$?


\[\mathbb{P}(X = k) = (1-p)^{k-1}p\]

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 $\mathbb{E}[X]$?


\[\frac{1}{p}\]

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 $\text{var}(X)$?


\[\frac{1-p}{p^2}\]

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 $G _ X(s)$?


\[\frac{ps}{1-(1-p)s}\]

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 $\mathbb{P}(X = k)$?


\[\mathbb{P}(X = k) = (1-p)^{k}p\]

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 $\mathbb{E}[X]$?


\[\frac{1-p}{p}\]

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 $\text{var}(X)$?


\[\frac{1-p}{p^2}\]

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 $G _ X(s)$?


\[\frac{p}{1-(1-p)s}\]

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

If $X$ is discrete and negatively binomially distributed, i.e. $X \sim \text{NegBin}(k, p)$, what is the probability mass function $\mathbb{P}(X = n)$?


\[\mathbb{P}(X = n) = \text{}{\text{}{n-1}\choose{k-1}\space} (1-p)^{n-k} p^k\]

If $X$ is discrete and negatively binomially distributed, i.e. $X \sim \text{NegBin}(k, p)$, what is the mean $\mathbb{E}[X]$?


\[\frac{k}{p}\]

If $X$ is discrete and negatively binomially distributed, i.e. $X \sim \text{NegBin}(k, p)$, what is the variance $\text{var}(X)$?


\[\frac{k(1-p)}{p^2}\]

If $X$ is discrete and negatively binomially distributed, i.e. $X \sim \text{NegBin}(k, p)$, what is the probability generation function $G _ X(s)$?


\[\left(\frac{ps}{1-(1-p)s}\right)^k\]

Continuous

Uniform

If $X$ is continuous and uniformly distributed, i.e. $X \sim U[a,b]$, what is the probability density function $f _ X(x)$?


\[\frac{1}{b-a}\]

If $X$ is continuous and uniformly distributed, i.e. $X \sim U[a,b]$, what is the cumulative distribution function $F _ X(x)$?


\[\frac{x-a}{b-a}\]

If $X$ is continuous and uniformly distributed, i.e. $X \sim U[a,b]$, what is the mean $\mathbb{E}[X]$?


\[\frac{a+b}{2}\]

If $X$ is continuous and uniformly distributed, i.e. $X \sim U[a,b]$, what is the variance $\text{var}(X)$?


\[\frac{(b-a)^2}{12}\]

Exponential

If $X$ is continuous and exponentially distributed, i.e. $X \sim \text{Exp}(\lambda)$, what is the probability density function $f _ X(x)$?


\[\begin{cases} \lambda e^{-\lambda x} &\text{if } x \ge 0 \\\\ 0 &\text{otherwise} \end{cases}\]

If $X$ is continuous and exponentially distributed, i.e. $X \sim \text{Exp}(\lambda)$, what is the cumulative distribution function $F _ X(x)$?


\[1 - e^{-\lambda x}\]

If $X$ is continuous and exponentially distributed, i.e. $X \sim \text{Exp}(\lambda)$, what is the mean $\mathbb{E}[X]$?


\[\frac{1}{\lambda}\]

If $X$ is continuous and exponentially distributed, i.e. $X \sim \text{Exp}(\lambda)$, what is the variance $\text{var}(X)$?


\[\frac{1}{\lambda^2}\]

Normal

If $X$ is continuous and normally distributed, i.e. $X \sim \text{N}(\mu, \sigma^2)$, what is the probability density function $f _ X(x)$?


\[\frac{1}{\sqrt{2\sigma^2\pi }\space} e^{-\frac{(x-\mu)^2}{2\sigma^2}\space}\]

If $X$ is continuous and normally distributed, i.e. $X \sim \text{N}(\mu, \sigma^2)$, what is the cumulative distribution function $F _ X(x)$ in terms of $\Phi(x)$?


\[\Phi\left(\frac{x-\mu}{\sigma}\right)\]



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