Probability MT22, Distributions

Created: January 05, 2023 | Updated: October 21, 2025 | Read markdown | About these notes | View in context | Study these flashcards |

In general

What’s another way of saying the mean of a distribution?

mean-is-expectation

The expectation of the distribution.

Discrete

Uniform

If $X$ is discrete and uniformly distributed, i.e. $X \sim U\{1, 2, \ldots, n\}$, what is the probability mass function $\mathbb{P}(X = k)$?

discrete-uniform-pmf
\[\mathbb{P}(X = k) = \frac{1}{n}\]

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

discrete-uniform-mean
\[\frac{n+1}{2}\]

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

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

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

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

Bernoulli

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

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

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

bernoulli-mean
\[p\]

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

bernoulli-variance
\[p(1-p)\]

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

bernoulli-pgf
\[1 - p + ps\]

Binomial

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

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

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

binomial-mean
\[np\]

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

binomial-variance
\[np(1-p)\]

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

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

Suppose that $X _ 1, \ldots, X _ n$ are independent $\mathrm{Ber}(p)$ random variables and let $Y = X _ 1 + \ldots + X _ n$. @Prove that $Y \sim \mathrm{Bin}(n, p)$.

sum-of-bernoullis-is-binomial-proof

@todo (probability, page 37).

Poisson

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

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

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

poisson-mean
\[\lambda\]

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

poisson-variance
\[\lambda\]

What is special about the Poisson distribution’s $\mathrm{Poi}(\lambda)$ mean and variance?

poisson-mean-equals-variance

They are the same, $\mathbb{E}[X] = \mathrm{var}(X) = \lambda$.

If $X$ is discrete and Poisson distributed, i.e. $X \sim \mathrm{Poi}(\lambda)$, what is the generating function $G _ X(s)$?

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

Suppose that $X _ 1, \ldots, X _ n$ are independent $\mathrm{Poi}(\lambda _ i)$ random variables and let $Y = X _ 1 + \ldots + X _ n$. @Prove that $Y \sim \mathrm{Poi}(\sum _ i \lambda _ i)$.

sum-of-independent-poissons-proof

@todo (probability, page 38).

Suppose that $X _ 1, \ldots$ are independent and identically distributed $\mathrm{Ber}(p)$ variables and that $N \sim \mathrm{Poi}(\lambda)$, independently of $X _ 1, \ldots$. @Prove that $\sum^N _ {i=1} X _ i \sim \mathrm{Poi}(\lambda p)$.

poisson-thinning-proof

@todo (probability, page 41).

Geometric ($k$ total)

If $X$ is discrete and geometrically distributed (in the case of $k$ total trials for the first success), i.e. $X \sim \mathrm{Geom}(p)$, what is the probability mass function $\mathbb{P}(X = k)$?

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

If $X$ is discrete and geometrically distributed (in the case of $k$ total trials for the first success), i.e. $X \sim \mathrm{Geom}(p)$, what is the mean $\mathbb{E}[X]$?

geometric-trials-mean
\[\frac{1}{p}\]

If $X$ is discrete and geometrically distributed (in the case of $k$ total trials for the first success), i.e. $X \sim \mathrm{Geom}(p)$, what is the variance $\mathrm{var}(X)$?

geometric-trials-variance
\[\frac{1-p}{p^2}\]

If $X$ is discrete and geometrically distributed (in the case of $k$ total trials for the first success), i.e. $X \sim \mathrm{Geom}(p)$, what is the generating function $G _ X(s)$?

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

Alternative Geometric ($k$ failures)

If $X$ is discrete and geometrically distributed (in the case of $k$ _ failures _ for the first success), i.e. $X \sim \mathrm{Geom}(p)$, what is the probability mass function $\mathbb{P}(X = k)$?

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

If $X$ is discrete and geometrically distributed (in the case of $k$ _ failures _ for the first success), i.e. $X \sim \mathrm{Geom}(p)$, what is the mean $\mathbb{E}[X]$?

geometric-failures-mean
\[\frac{1-p}{p}\]

If $X$ is discrete and geometrically distributed (in the case of $k$ _ failures _ for the first success), i.e. $X \sim \mathrm{Geom}(p)$, what is the variance $\mathrm{var}(X)$?

geometric-failures-variance
\[\frac{1-p}{p^2}\]

If $X$ is discrete and geometrically distributed (in the case of $k$ _ failures _ for the first success), i.e. $X \sim \mathrm{Geom}(p)$, what is the generating function $G _ X(s)$?

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

Negative Binomial

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

negative-binomial-interpretation

The probability that the $k$-th success occurs on trial $n$ (i.e. $X$ counts the total number of trials needed for $k$ successes).

Why are the expressions for the expectation and variance of $X \sim \mathrm{NegBin}(k, p)$ equal to $k$ times the expectation and variance of a geometrically distributed random variable?

negative-binomial-vs-geometric

Because the negative 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-vs-negative-binomial
  • Binomial: Models the value of a sum of Bernoulli variables
  • Negative binomial: Models the number of Bernoulli trials needed for a certain number of successes

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

negative-binomial-pmf
\[\mathbb{P}(X = n) = \binom{n-1}{k-1} (1-p)^{n-k} p^k\]

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

negative-binomial-mean
\[\frac{k}{p}\]

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

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

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

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

Continuous

Uniform

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

continuous-uniform-pdf
\[\frac{1}{b-a}\]

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

continuous-uniform-cdf
\[\frac{x-a}{b-a}\]

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

continuous-uniform-mean
\[\frac{a+b}{2}\]

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

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

Exponential

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

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

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

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

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

exponential-mean
\[\frac{1}{\lambda}\]

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

exponential-variance
\[\frac{1}{\lambda^2}\]

Normal

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

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

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

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

What is the notation for the CDF of the standard normal distribution, $F _ Z(z)$?

standard-normal-cdf-notation
\[\Phi(z)\]

What, in terms of an integral, is the formula for the CDF of the standard normal distribution $\Phi(z)$?

standard-normal-cdf-integral
\[\Phi(z) = \int^z _ {-\infty} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \, dx\]

If $X$ and $Y$ are jointly normally distributed random variables, what does $\mathrm{Cov}(X, Y) = 0$ imply (and why do you need to be careful)?

jointly-normal-zero-covariance
\[\mathrm{Cov}(X, Y) = 0 \iff X, Y \text{ independent}\]

Need to be careful since this equivalence fails for general (non-jointly-normal) random variables.



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