Monthly Archives: May 2008

Conditional Probability and Expectation

Conditional probability: $\Pr(X\mid Y) = \displaystyle \frac{\Pr(X \cap Y)}{\Pr(Y)}$

Bayes Theorem: $\Pr(A\mid B) = \displaystyle \frac{\Pr(B \mid A)\Pr(A)}{\Pr(B)}$

for continuous distributions: $f_X(x\mid y) = \displaystyle \frac{f_Y(y \mid x)f_X(x)}{f_Y(y)}$

Recall for a joint distribution function $f(x,y)$, $f_X(x) = \displaystyle \int_{-\infty}^\infty {f(x,y)dy}$

Law of Total Probability:  Suppose $\displaystyle \sum_{i=1}^n B_i = 1$ and $\Pr(B_i \cap B_j) = 0$ for $i \ne j$, then for any event $A$, $\begin{array}{rl} \Pr(A) &= \displaystyle \sum_{i=1}^n \Pr(A \cap B_i) \\ &= \displaystyle \sum_{i=1}^n \Pr(B_i)\Pr(A\mid B_i) \end{array}$

In many cases, you will need to use the law of total probability in conjunction with Bayes Theorem to find $P(A)$ or $P(B)$.

For a continuous distribution: $\Pr(A) = \displaystyle \int\Pr(A\mid x)f(x)dx$

Conditional Mean: $E_X[X] = E_Y[E_X[X\mid Y]]$

Percentiles

For a random variable $X$, the 100pth percentile is the unique value $\pi_p$ such that $\Pr(X \le \pi_p)=p$.  A qth quantile is a value $\pi_q$ such that $\Pr(X \le \pi_q)=q$.  For example, if $F(1120) = 0.2$, we can either say 1120 is the twentieth percentile or 1120 is the one-fifth quantile.

Filed under Probability

Functions and Moments

Some distribution functions:

Survival function $\displaystyle S(x) = 1-F(x) = \Pr(X>x)$

where $F(x)$ is a cumulative distribution function.

Hazard rate function $\displaystyle h(x) = \frac{f(x)}{S(x)} = -\frac{d\ln{S(x)}}{dx}$

where $f(x)$ is a probability density function.

Cumulative hazard rate function $\displaystyle H(x) =\int_{-\infty}^x{h(t)dt} = -\ln{S(x)}$

The following relationship is often useful: $S(x) = \displaystyle e^{-\int_{-\infty}^x{h(t)dt}}$

Expected Value: $\displaystyle E[X] = \int_{-\infty}^\infty{xf(x)dx}$

Or more generally, $\displaystyle E[g(X)] = \int_{-\infty}^\infty{g(x)f(x)dx}$

When $g(X) = X^n$, the expected value of such a function is called the nth raw moment and is denoted by $\mu'_n$.  Let $\mu$ be the first raw moment.  That is, $\mu = E[X]$. $E[(X-\mu)^n]$ is called an nth central moment.

Moments are used to generate some statistical measures.

Variance $\sigma^2$ $\displaystyle Var(X) = E[(X-\mu)^2] = E(X^2) - E(X)^2$

The coefficient of variation is $\displaystyle \frac{\mu}{\sigma}$.

Skewness $\gamma_1$ $\displaystyle \gamma_1 = \frac{\mu_3}{\sigma^3}$

Kurtosis $\gamma_2$ $\displaystyle \gamma_2 = \frac{\mu_4}{\sigma^4}$

Covariance of two distribution functions $\displaystyle Cov(X,Y) = E[(X-\mu_x)(Y-\mu_Y)] = E[XY] - E[X]E[Y]$

*Note: if $X$ and $Y$ are independent, $Cov(X,Y)=0$

Correlation coefficient $\rho_{XY}$ $\displaystyle \rho_{XY} = \frac{Cov(X,Y)}{\sigma_X\sigma_Y}$

All of the above definitions should be memorized.  Some things that might be tested in the exam are:

• Given a particular distribution function, what happens to skewness or kurtosis in the limit of a certain parameter?
• What is the expected value, variance, skewness, kurtosis of a given distribution function?
• What is the covariance or correlation coefficient of two distribution functions?

Central moments can be calculated using raw moments.  Know how to calculate raw moments using the statistics function on the calculator.  This can be a useful timesaver in the exam.  Using alternating positive and negative binomial coefficients, write an expression for $\mu_n$ with $\mu'$ and $\mu$ as the two binomial terms.

Example: $\mu_4 = \mu'_4 - 4\mu'_3\mu + 6\mu'_2\mu^2 - 4\mu'_1\mu^3 + \mu^4$

Since $\mu'_1 = \mu$, the two terms on the end simplify to $-3\mu^4$.  The result is $\mu_4 = \mu'_4 - 4\mu'_3\mu + 6\mu'_2\mu^2 - 3\mu^4$

Moment Generating Function:

If the moment generating function $M(t)$ is known for random variable $X$, it’s nth raw moment can be found by taking the nth derivative of $M(t)$ and evaluating at 0.  Moment generating functions take the form: $M(t) = \displaystyle E[e^{tx}]$

If $Z = X +Y$, then $M_Z(t) = M_X(t)\cdot M_Y(t)$.

Goals, Purpose, Procedures, Etc.

Effective study requires one to have clear goals in mind and to have a system by which to measure progress. Efficient study requires focus, attention and diligence throughout an entire study session. Complete coverage of all topics required for an exam takes dedication and hard work.

My study regiment so far has had none of these qualities.   The goal of this blog will be to document my study behaviors in order to become a better actuarial student and increase my pass ratio.

The measurement of progress will be based on a point system as follows:

• 5 pts. — Completing a blog summary of a study session (I’m so lazy that this requires an award of points to serve as motivation)
• 5 pts. — Reading a page of the study manual
• 5 pts. — Reading a page of the textbook
• 10 pts. — Completing a study manual problem
• 15 pts. — Completing a practice exam problem
• 200 pts. — Completing a timed sample exam

My goal will be to try to achieve 50 to 100 pts. on weekdays and 100 to 150 pts. on weekends and holidays. The amount of study time per session will also be logged as a way to measure efficiency in pts/hr.

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