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

24 sample cards from the Exam P — Probability deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

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

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

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

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

3. 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 $\dfrac{(b-a)^{2}}{12}$.

4. 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: $P(X > s + t \mid X > s) = P(X > t)$.

5. 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 $\alpha\theta^{2}$.

6. 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 $\sigma^{2}$.

7. 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.

8. 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$(n, 1/\lambda)$.

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

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

10. 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$(\lambda)$.

11. 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$(\lambda)$.

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

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

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

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

14. 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).

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

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

16. 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 $-1 \leq \rho \leq 1$.

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

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

18. 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.

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

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

20. State the law of total variance.

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

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

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

22. 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 $\sum_{i=1}^{n} X_{i} \approx N(n\mu, n\sigma^{2})$.

23. 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. $P(X \leq k) \approx P\!\left(Z \leq \dfrac{k + 0.5 - \mu}{\sigma}\right)$.

24. 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).

What this deck covers

Exam P — Probability flashcards FAQ