Associate of the Society of Actuaries (ASA/FSA) Exam P — Probability Syllabus

How is the cumulative distribution function $F(x)$ related to the density $f(x)$ for a continuous random variable?

$F(x) = \int_{-\infty}^{x} f(t)\,dt$ and $f(x) = F'(x)$ where the derivative exists.

Express $E[X]$ and $E[g(X)]$ for a continuous random variable.

$E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx$ and $E[g(X)] = \int_{-\infty}^{\infty} g(x)\,f(x)\,dx$.

For a nonnegative continuous random variable, give the survival-function formula for the mean.

$E[X] = \int_{0}^{\infty} \bigl(1 - F(x)\bigr)\,dx = \int_{0}^{\infty} S(x)\,dx$

Define the moment generating function (MGF) of a random variable $X$.

$M_{X}(t) = E\!\left[e^{tX}\right]$, for $t$ in a neighborhood of $0$.

How do you obtain the $n$-th moment of $X$ from its MGF?

$E[X^{n}] = M_{X}^{(n)}(0)$, the $n$-th derivative of $M_{X}(t)$ evaluated at $t = 0$.

If $X$ and $Y$ are independent, what is the MGF of $X + Y$?

$M_{X+Y}(t) = M_{X}(t)\,M_{Y}(t)$

What is the MGF of the linear transformation $Y = aX + b$ in terms of $M_{X}$?

$M_{Y}(t) = e^{bt} M_{X}(at)$

Describe the CDF method for finding the distribution of $Y = g(X)$.

Compute $F_{Y}(y) = P(g(X) \leq y)$ by expressing the event in terms of $X$, then differentiate: $f_{Y}(y) = F_{Y}'(y)$.

State the one-to-one transformation (change-of-variable) formula for the density of $Y = g(X)$.

$f_{Y}(y) = f_{X}\!\bigl(g^{-1}(y)\bigr)\left|\dfrac{d}{dy}g^{-1}(y)\right|$

Give the PMF, mean, and variance of a Binomial$(n, p)$ random variable.

$p(k) = \binom{n}{k} p^{k}(1-p)^{n-k}$, mean $np$, variance $np(1-p)$.

Give the PMF, mean, and variance of a Poisson$(\lambda)$ random variable.

$p(k) = \dfrac{e^{-\lambda}\lambda^{k}}{k!}$, mean $\lambda$, variance $\lambda$.

Give the PMF, mean, and variance of a Geometric random variable (number of trials until first success).

$p(k) = (1-p)^{k-1}p$ for $k = 1, 2, \dots$, mean $\dfrac{1}{p}$, variance $\dfrac{1-p}{p^{2}}$.

Give the mean and variance of a Negative Binomial random variable counting trials to the $r$-th success.

Mean $\dfrac{r}{p}$, variance $\dfrac{r(1-p)}{p^{2}}$.

Give the mean and variance of a Hypergeometric distribution sampling $n$ from $N$ items with $K$ successes.

Mean $n\dfrac{K}{N}$, variance $n\dfrac{K}{N}\dfrac{N-K}{N}\dfrac{N-n}{N-1}$.

Give the density, mean, and variance of a Uniform$(a, b)$ random variable.

$f(x) = \dfrac{1}{b-a}$ for $a \leq x \leq b$, mean $\dfrac{a+b}{2}$, variance $\dfrac{(b-a)^{2}}{12}$.

Give the density, mean, variance, and memoryless property of an Exponential$(\lambda)$ random variable.

$f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$, mean $\dfrac{1}{\lambda}$, variance $\dfrac{1}{\lambda^{2}}$; memoryless: $P(X > s + t \mid X > s) = P(X > t)$.

Give the density, mean, and variance of a Gamma$(\alpha, \theta)$ random variable (scale parameterization).

$f(x) = \dfrac{x^{\alpha-1}e^{-x/\theta}}{\Gamma(\alpha)\theta^{\alpha}}$ for $x > 0$, mean $\alpha\theta$, variance $\alpha\theta^{2}$.

Give the density, mean, and variance of a Normal$(\mu, \sigma^{2})$ random variable.

$f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\dfrac{(x-\mu)^{2}}{2\sigma^{2}}\right)$, mean $\mu$, variance $\sigma^{2}$.

How do you standardize a Normal random variable $X \sim N(\mu, \sigma^{2})$?

$Z = \dfrac{X - \mu}{\sigma} \sim N(0, 1)$, the standard normal.

What distribution arises as the sum of $n$ independent Exponential$(\lambda)$ random variables?

A Gamma distribution with shape $n$ and rate $\lambda$ (an Erlang distribution): Gamma$(n, 1/\lambda)$.

How is the Exponential distribution a special case of the Gamma distribution?

Exponential$(\lambda)$ is Gamma with shape parameter $\alpha = 1$ and rate $\lambda$.

What is the relationship between the Poisson and Exponential distributions in a Poisson process?

If events occur per a Poisson process with rate $\lambda$ (counts are Poisson$(\lambda t)$), the waiting times between events are independent Exponential$(\lambda)$.

By the limit relationship, when does a Binomial$(n, p)$ approximate a Poisson, and with what parameter?

When $n$ is large and $p$ is small with $np = \lambda$ moderate, Binomial$(n, p) \approx$ Poisson$(\lambda)$.

How do you obtain marginal densities from a joint density $f(x, y)$?

$f_{X}(x) = \int_{-\infty}^{\infty} f(x, y)\,dy$ and $f_{Y}(y) = \int_{-\infty}^{\infty} f(x, y)\,dx$.

Define the conditional density of $Y$ given $X = x$.

$f_{Y \mid X}(y \mid x) = \dfrac{f(x, y)}{f_{X}(x)}$, provided $f_{X}(x) > 0$.

What condition on the joint density characterizes independence of continuous $X$ and $Y$?

$f(x, y) = f_{X}(x)\,f_{Y}(y)$ for all $x, y$ (the support must also factor as a product).

Give the two equivalent formulas for the covariance of $X$ and $Y$.

$\operatorname{Cov}(X, Y) = E[(X - \mu_{X})(Y - \mu_{Y})] = E[XY] - E[X]E[Y]$

Define the correlation coefficient $\rho$, and state its range.

$\rho_{X,Y} = \dfrac{\operatorname{Cov}(X, Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}}$, with $-1 \leq \rho \leq 1$.

What is the variance of a sum $\operatorname{Var}(aX + bY)$ in terms of covariance?

$\operatorname{Var}(aX + bY) = a^{2}\operatorname{Var}(X) + b^{2}\operatorname{Var}(Y) + 2ab\operatorname{Cov}(X, Y)$

If $X$ and $Y$ are independent, what are $\operatorname{Cov}(X, Y)$ and $\rho$? Is the converse true?

$\operatorname{Cov}(X, Y) = 0$ and $\rho = 0$. The converse is not generally true: uncorrelated does not imply independent.

State the double-expectation (tower) property for conditional expectation.

$E[X] = E\bigl[E[X \mid Y]\bigr]$

State the law of total variance.

$\operatorname{Var}(X) = E\bigl[\operatorname{Var}(X \mid Y)\bigr] + \operatorname{Var}\bigl(E[X \mid Y]\bigr)$

Define the conditional expectation $E[X \mid Y = y]$ in the continuous case.

$E[X \mid Y = y] = \int_{-\infty}^{\infty} x\,f_{X \mid Y}(x \mid y)\,dx$

State the Central Limit Theorem for the sample mean $\bar{X}_{n}$ of i.i.d. variables with mean $\mu$ and variance $\sigma^{2}$.

As $n \to \infty$, $\dfrac{\bar{X}_{n} - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0, 1)$; equivalently $\sum_{i=1}^{n} X_{i} \approx N(n\mu, n\sigma^{2})$.

What continuity correction is applied when approximating a discrete (e.g. Binomial) count with the Normal distribution?

Adjust the boundary by $\pm 0.5$: e.g. $P(X \leq k) \approx P\!\left(Z \leq \dfrac{k + 0.5 - \mu}{\sigma}\right)$.

State the Law of Large Numbers for the sample mean.

As $n \to \infty$, the sample mean $\bar{X}_{n}$ converges to the population mean $\mu$ (in probability for the weak law).