 Actuarial Exam P

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Pelham Delaney (lvl 10)
Algebra & Calculus Review

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Algebra & Calculus Review

(42 cards)

$$lne^{x} =$$

Front

$$=x$$

Back

$$e^{x\ln b} =$$

Front

$$=b^x$$

Back

Double Integral

Given a continuous function $$f(x,y)$$ on the rectangular region bounded by $$x=a$$, $$x=b$$, $$y=c$$, and $$y=d$$, what is the definite integral of  $$f$$?

Front

Can be expressed in two ways:

$$\int_{a}^{b} \int_{c}^{d} f(x,y) \,dy\,dx$$

$$\int_{c}^{d} \int_{a}^{b} f(x,y) \,dx\,dy$$

Back

$$\int a^x \,dx$$

Front

$$\frac{a^x}{lna}$$

Back

Derivative of $$a^x$$

Front

$$a^x \cdot lna$$

Back

Derivative of $$e^{g(x)}$$

Front

$$g'(x)\cdot e^{g(x)}$$

Back

Increasing geometric series sum

$$1 + 2r + 3r^2 + \cdots =$$

Front

$$\frac{1}{(1-r)^2}$$

Back

Derivative of $$cosx$$

Front

$$-sinx$$

Back

$$\sum_{x=0}^{\infty} \frac{a^x}{x!} =$$

Front

$$= e^a$$

Back

Derivative of $$e^x$$

Front

$$e^x$$

Back

$$b^{\log _{b} y} =$$

Front

$$=y$$

Back

Arithmetic Progression

Sum of the first $$n$$ terms of the series $$a + (a + d) + (a + 2d) +\cdots$$

Front

$$na + d \cdot \frac{n(n-1)}{2}$$

Back

Integration by Parts

A technique of integration based on the product rule to find $$\int f(x) \cdot g'(x) \,dx$$

Front

$$\int f(x) \cdot g'(x) \,dx = f(x) \cdot g(x) - \int f'(x) \cdot g(x) \,dx$$

Useful if $$f'(x) \cdot g(x)$$ has an easier antiderivative to find than $$f(x) \cdot g'(x)$$. May be necessary to apply integration by parts more than once to simplify an integral.

Back

One-to-one function

Front

A function where each element of the range is paired with exactly one element of the domain (only one y value for each x value)

Back

Partial Differentiation

Given function $$f(x,y)$$, what is the partial derivative of $$f$$ with respect to $$x$$ at the point $$(x_{0},y_{0})$$?

Front

$$\frac{\partial f}{\partial x}$$ is found by differentiating $$f$$ with respect to $$x$$ and regarding $$y$$ as constant, then substituting the values $$x = x_{0}$$ and $$y = y_{0}$$

Back

if $$G(x) = \int_{a}^{h(x)} f(u) \,du$$

$$G'(x) =$$

Front

$$= f[h(x)]\cdot h'(x)$$

Back

Product rule

(derivative of $$g(x)\cdot h(x)$$

Front

$$g'(x)\cdot h(x) + g(x)\cdot h'(x)$$

Back

$$\int sinx \,dx$$

Front

$$-cosx$$

Back

Inverse of exponential function $$f(x) = b^x$$

Front

$$x = log _{b} y$$

Back

Integration of $$f$$ on $$a,b$$ when $$f$$ is not defined at $$a$$ or $$b$$ or when $$a$$ or $$b$$ is $$\pm \infty$$

Front

Integration over an infinite interval is defined by taking limits $$\int_{a}^{\infty} f(x) \,dx = \lim\limits_{b \to \infty} \int_{a}^{b} f(x) \,dx$$

If $$f$$ is not defined or is discontinuous at $$x = a$$ then $$\int_{a}^{b} f(x) \,dx = \lim\limits_{c \to a+} \int_{c}^{b} f(x) \,dx$$

If $$f$$ is undefined or discontinuous at point $$x = c$$ then $$\int_{a}^{b} f(x) \,dx = \int_{a}^{c} f(x) \,dx + \int_{c}^{b} f(x) \,dx$$

Back

Quotient Rule

(derivative of $$\frac{g(x)}{h(x)}$$

Front

$$\frac{h(x)g'(x)-g(x)h'(x)}{[h(x)]^2}$$

Back

$$\int cosx \,dx$$

Front

$$sinx$$

Back

$$\int_{0}^{\infty} x^{n}e^{-cx} \,dx =$$

Front

$$= \frac{n!}{c^{n+1}}$$

Back

L'Hospital's First Rule

For limits of the form $$\lim\limits_{x \to c} \frac{f(x)}{g(x)}$$

IF

(i) $$\lim\limits_{x \to c} f(x) = \lim\limits_{x \to c} g(x) = 0$$ and

(ii) $$f'(c)$$ exists and

(iii) $$g'(c)$$ exists and $$\neq$$ 0

(note that 0 can be replaced with $$\pm \infty$$)

THEN

Front

$$\lim\limits_{x \to c} \frac{f(x)}{g(x)} = \frac{f'(c)}{g'(c)}$$

Back

Derivative of $$lnx$$

Front

$$\frac{1}{x}$$

Back

$$e^{lny}=$$

Front

$$=y$$

Back

Derivative of $$ln(g(x))$$

Front

$$\frac{g'(x)}{g(x)}$$

Back

if $$G(x) = \int_{g(x)}^{h(x)} f(u) \,du$$

$$G'(x) =$$

Front

$$G'(x) = f[h(x)]\cdot h'(x) - f[g(x)]\cdot g'(x)$$

Back

Integration by Substitution

To find if $$\int f(x) \,dx$$ we make the substitution $$u = g(x)$$ for an appropriate function $$g(x)$$ which has an antiderivative easier to find than the original.

Front

The differential $$du$$ is defined as $$du = g'(x)dx$$ and we try to write $$\int f(x) \dx$$ as an integral with respect to the variable $$u$$.

For example, to find $$\int(x^3-1)^{4/3}x^2 \,dx$$, we let $$u = x^3 - 1$$ so that $$du = 3x^2dx$$ or equivalently, $$\frac{1}{3}\cdot du = x^2dx$$ and the integral can be rewritten as $$\int u^{\frac{4}{3}}\cdot \frac{1}{3} \,du$$

Back

if $$G(x) = \int_{g(x)}^{b} f(u) \,du$$

$$G'(x) =$$

Front

$$G'(x) = -f[g(x)\cdot g'(x)$$

Back

Front

$$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Back

Chain Rule

(derivative of $$g(h(x))$$

Front

$$g'(h(x)) \cdot h'(x)$$

Back

$$\int xe^{ax} \,dx$$

Front

$$\frac{xe^{ax}}{a} - \frac{e^{ax}}{a^2}$$

Back

$$\log _{b} \frac{y}{z} =$$

Front

$$= \log _{b} y - \log _{b} z$$

Back

Derivative of $$\log _{b} x$$

Front

$$\frac{1}{xlnb}$$

Back

Geometric Progression

Sum of the first $$n$$ terms $$a+ar+ar^2+\cdots + ar^{n-1}$$

Infinite series sum $$a+ar+ar^2+\cdots$$

Front

Sum of the first $$n$$ terms $$= a\cdot \frac{r^n-1}{r-1}$$

Infinite series sum $$= \frac{a}{1-r}$$

Back

L'Hospital's Second Rule

For limits of the form $$\lim\limits_{x \to c} \frac{f(x)}{g(x)}$$

IF

(i) $$\lim\limits_{x \to c} f(x) = \lim\limits_{x \to c} g(x) = 0$$ and

(ii) $$f$$ and $$g$$ are differentiable near $$c$$ and

(iii) $$\lim\limits_{x \to c} \frac{f'(x)}{g'(x)}$$ exists

(note that 0 can be replaced with $$\pm \infty$$)

THEN

Front

$$\lim\limits_{x \to c} \frac{f(x)}{g(x)} = \lim\limits_{x \to c} \frac{f'(x)}{g'(x)}$$

Back

$$log _{b} y =$$

Front

$$= \frac{lny}{lnb}$$

Back

Power rule

(derivative of $$cx^n$$)

Front

$$cnx^{n-1}$$

Back

$$\int \frac{1}{x} \,dx$$

Front

$$lnx$$

Back

Derivative of $$sinx$$

Front

$$cosx$$

Back

$$\log _{b} yz =$$

Front

$$\log _{b} y + \log _{b} z$$

Back

Basic Probability

(9 cards)

$$(A\cup B)' =$$

Front

$$(A\cup B)' = A'\cap B'$$

Back

Mutually exclusive events

Front

Cannot occur simultaneously. No sample points in common. Also referred to as disjoint or as having empty intersection.

Back

Event

Front

A collection of sample points, a subset of the probability space. We say "event A has occurred" if the experimental outcome was one of the sample points in A.

Back

Sample Point and Sample Space

Front

A sample point is the simple outcome of a random experiment. The sample space or probability space is the collection of all possible sample points (outcomes) related to a specified experiment.

Back

$$A\cup (B_{1}\cap B_{2}\cap \cdots B_{n}) =$$

Front

$$A\cup (B_{1}\cap B_{2}\cap \cdots B_{n}) = (A\cup B_{1})\cap (A\cup B_{2})\cap \cdots \cap (A\cup B_{n})$$

Back

Complement of event A

Front

All sample points in the probability space that are not in A.

A'

Back

$$A\cap (B_{1}\cup B_{2}\cup \cdots B_{n}) =$$

Front

$$A\cap (B_{1}\cup B_{2}\cup \cdots B_{n}) = (A\cap B_{1})\cup (A\cap B_{2})\cup \cdots (A\cap B_{n})$$

Back

Partition of event A

Front

Events C1, C2,...,Cn form a partition of event A if the Cs are mutually exclusive and exhaustive of event A.

Back

$$(A\cap B)' =$$

Front

$$(A\cap B)' = A'\cup B'$$

Back

Conditional Probability and Independence

(9 cards)

$$P[B|A] =$$

Front

$$\frac{P[B\cap A]}{P[A]}$$

Back

Bayes' rule and Theorem

For events A and B, P[A|B] =

(usually used to turn around conditioning of events A and B)

Front

$$P[A|B] = \frac{P[A\cap B]}{P[B\cap A] + P[B\cap A']} = \frac{P[B|A]\cdot P[A]}{P[B|A]\cdot P[A] + P[B|A']\cdot P[A']}$$

Back

Relationship between independent events A and B

Front

$$P[A\cap B] = P[A]\cdot P[B]$$

Back

Law of Total Probability

For events A and B, P[B] =

Front

$$P[B] = P[B|A]\cdot P(A) + P[B|A']\cdot P(A')$$

Back

$$P[A'|B] =$$

Front

$$1 - P[A|B]$$

Back

$$P[A\cup B|C] =$$

Front

$$P[A\cup B|C] = P[A|C] + P[B|C] - P[A\cap B|C]$$

Back

$$P[A\cup B|C] =$$

Front

$$P[A|C] + P[B|C] - P[A\cap B|C]$$

Back

$$P[B\cap A] =$$

Front

$$P[B|A]\cdot P[A]$$

Back

$$P[A_{1} \cap A_{2}\cap \cdots \cap A_{n}] =$$

Front

$$P[A_{1}]\cdot P[A_{2}|A_{1}]\cdot P[A_{3}|A_{1}\cap A_{2}]\cdots P[A_{n}|A_{1}\cap A_{2}\cap \cdots \cap A_{n-1}]$$

Back

Combinatorial Principles, Permutations and Combinations

(6 cards)

Multinomial Theorem

In the power series expansion of $$(t_{1} + t_{2} +\cdots + t_{s} )^N$$ the coefficient of $$t_{1}^{k_{1}}\cdot t_{2}^{k_{2}}\cdots t_{s}^{k_{s}}$$ is

Front

$$\binom{N}{k_{1}, k_{2}, \cdots k_{s}} = \frac{N!}{k_{1}!\cdot k_{2}! \cdots k_{s}!}$$

For example, in the expansion of $$(1+x+y)^4$$, the coefficient of $$xy^2$$ is the coefficient of $$1^{1}x^{1}y^{2}$$ which is $$\frac{4!}{1!\cdot 1!\cdot 2!} = 12$$

Back

Number of ordering $$n$$ objects with $$n_{1}$$ of Type 1,  $$n_{2}$$ of Type 2, ...,  $$n_{t}$$ of Type $$t$$

Front

$$\frac{n_{1}}{n_{1}!\cdot n_{2}!\cdots n_{t}!}$$

Back

Binomial Theorem

Power series expansion of $$(1+t)^N$$

Front

The coefficient of $$t^k$$ is $$\binom{N}{k}$$ so that $$(1+t)^N = \sum_{k=0}^{\infty}\binom{N}{k}\cdot t^k = 1 + Nt + \frac{N(N-1)}{2}t^2 + \frac{N(N-1)(N-2)}{6}t^2 \cdots$$

Back

Way of choosing a subset size $$k$$ of $$n$$ objects with $$k_{1}$$ objects of Type 1, $$k_{2}$$ objects of Type 2, ..., and $$k_{t}$$ objects of Type $$t$$

Front

$$\binom{n_{1}}{k_{1}}\cdot \binom{n_{2}}{k_{2}}\cdots \binom{n_{t}}{k_{t}}$$

Back

Number of permutations of size $$k$$ out of $$n$$ distinct objects

Front

Denoted $$_{n}P_{k}$$

$$\frac{n!}{(n-k)!}$$

Back

Number of combinations of size $$k$$ out of $$n$$ distinct objects

Front

Denoted $$\binom{n}{k}$$ or $$_{n}C_{k}$$

$$\frac{n!}{k!\cdot (n-k)!}$$

Back

Random Variables and Probability Distributions

(5 cards)

Probability function of a discrete random variable

Front

Usually denoted $$p(x), f_{X},$$ or $$p_{x}$$

The probability that the value $$x$$ occurs

(i) $$0 ≤ p(x) ≤1$$ for all $$x$$, and

(ii) $$\sum_{x} p(x) = 1 Back Survival function \(S(x)$$

Front

The complement of the cumulative distribution function

$$S(x) = 1 - F(X) = P[X > x]$$

Back

Probability density function (pdf)

Front

Usually denoted $$f(x)$$ or $$f_{X} (x)$$

Probabilities related to X are found by integrating the density function over an interval.

$$P[X\in (a,b)] = P[a < X < b] = \int_{a}^{b} f(x) dx$$

Must satisfy:

(i) $$f(x) ≥ 0$$ for all $$x$$, and

(ii) $$\int_{-\infty}^{\infty} f(x) dx = 1$$

Back

Cumulative distribution function $$F(x)$$ or $$F_{X} (x)$$

Front

$$F(x) = P[X ≤ x]$$

For a discrete random variable, $$F(x) = \sum_{w<x} p(w)$$

For a continuous random variable, $$F(x) = \int_{-\infty}^{\infty} f(t)dt$$ and $$F(x)$$ is a continuous, differentiable, non-decreasing function.

Back

Expectation and Other Distribution Parameters

(23 cards)

Finding $$Var[X]$$ using $$M_{X}(t)$$ and logarithms

Front

$$Var[X] = \frac{d^2}{dt^2} ln[M_{X}(t)]|_{t=0}$$

Back

$$E[aX+b]$$

Front

$$= aE[X]+b$$

Back

Skewness of a distribution

Front

$$\frac{E[(X-μ)^3]}{σ^3}$$

If the skewness is positive, the distribution is said to be skewed to the right. If it is negative, it is skewed to the left.

Back

Standard Deviation of $$X$$

$$σ_{X}$$

Front

$$= \sqrt{Var[X]}$$

Back

Finding $$E[X]$$ using $$M_{X}(t)$$ and logarithms

Front

$$E[X] = \frac{d}{dt} ln[M_{X}(t)]|_{t=0}$$

Back

A mixture of distributions - expected values and moment generating functions for density function $$f(x) = a_{1}f_{1}(x) + a_{2}f_{2}(x) + \cdots a_{k}f_{k}(x)$$

Front

$$E[X^{n}] = a_{1}E[X_{1}^{n}] + a_{2}E[X_{2}^{n}] + \cdots a_{k}E[X_{k}^{n}]$$

$$M_{x}(t) = a_{1}M_{X_1}(t) + a_{2}M_{X_2}(t) + \cdots + a_{k}M_{X_k}(t)$$

Back

If X is a random variable defined on the interval $$[a,b]$$, $$E[X] =$$

Front

$$E[X] = a + \int_{a}^{b}[1 - F(x)]dx$$

Back

$$n$$-th moment of $$X$$

Front

$$E[X^n]$$

Back

Coefficient of variation of X

Front

$$\frac{σ_{x}}{μ_{x}}$$

Back

$$Var[aX+b]$$

Front

$$Var[aX+b] = a^2 Var[X]$$

Back

Median of distribution $$X$$

Front

The point $$M$$ for which $$P[X ≤ M] = .5$$

Back

Taylor expansion $$e^y =$$

Front

$$e^y = 1 + y + \frac{y^2}{2!} + \frac{y^3}{3!} + \cdots$$

Back

Taylor series expansion of $$M_{x}(t)$$ about point $$t=0$$

Front

$$M_{x}(t) = \sum_{k=0}^{\infty} \frac{t^k}{k!} E[X^k] = 1 + t\cdot E[X] + \frac{t^2}{2!} \cdot E[X^2] + \frac{t^3}{3!} \cdot E[X^3] \cdots$$

For a discrete distribution with probability function $$p_{k}$$,

$$M_{x}(t) = e^{tx_1}\cdot p_{1} + e^{tx_2}\cdot p_{2} + e^{tx_3}\cdot p_{3} \cdots$$

Back

Chebyshev's inequality

For random variable $$X$$ with mean $$μ_{x}$$ and standard deviation $$σ_{x}$$

For any real number $$r > 0$$...

Front

$$P[|X - μ_{x}| > rσ_{x}] ≤ \frac{1}{r^2}$$

Back

Mode of a distribution

Front

Any point $$m$$ at which the probability or density function $$f(x)$$ is maximized.

Back

$$n$$-th central moment of $$X$$ about the mean $$μ$$

Front

$$E[(X-μ)^n]$$

Back

Percentiles of distribution

Front

For $$0 < p < 1$$, the 100$$p$$-th percentile of the distribution of $$X$$ is the number $$c_{p}$$ which satisfies:

$$P[X ≤ c_{p}] ≥ p$$

$$P[X ≥ c_{p}] ≤ 1 - p$$

Back

For constants $$a_{1}, a_{2}$$, and $$b$$ and functions $$h_{1}$$ and $$h_{2}$$,

$$E[a_{1}h_{1}(X) + a_{2}h_{2}(X) + b]$$

Front

$$E[a_{1}h_{1}(X) + a_{2}h_{2}(X) + b] = a_{1}E[h_{1}(X)] + a_{2}E[h_{2}(X)] + b$$

Back

Expected value of a random variable X

$$E[X], μ_{x}$$

Front

For a discrete random variable:

$$E[X] = \sum x\cdot p(x) = x_{1}\cdot p(x_{1}) + x_{2}\cdot p(x_{2}) + \cdots$$

For a continuous random variable:

$$E[X] = \int_{-\infty}^{\infty} x\cdot f(x)dx$$

(Interval of integration is the interval of non-zero density for $$X$$)

Back

Expected value of function $$h(x)$$

$$E[h(x)]$$

Front

For a discrete random variable:

$$E[h(x)] = \sum_{x} h(x)\cdot p(x)$$

For a continuous random variable with density function $$f(x)$$:

$$E[h(x)] = \int_{-\infty}^{\infty} h(x)\cdot f(x)dx$$

Back

Jensen's Inequality

For function $$h$$ and random variable $$X$$

Front

If $$h''(x) ≥ 0$$ at all points $$x$$ with non-zero probability for $$X$$, then $$E[h(X)] ≥ h(E[X])$$ and if $$h''(x) > 0$$ at all points $$x$$ with non-zero probability for $$X$$, then $$E[h(X)] > h(E[X])$$

The inequality reverses for $$h''(x) ≤ 0$$

Back

Moment generating function of $$X$$

$$M_{X}(t)$$

Front

$$M_{X}(t) = E[e^{tX}]$$

(i) It is always true that $$M_{X}(0) = 1$$

(ii) Moments of X can be found by successive derivations of $$M_{X}(t)$$, e.g. $$M_{X}'(0) = E[X]$$, $$M_{X}''(0) = E[X^2]$$, etc.

Back

$$Var[X]$$

$$σ^2$$ or $$σ_{x}^2$$

Front

$$= E[(X-μx)^2] = E[X^2] - (E[X])^2 = E[X^2] - μ_{x}^2$$

Back

Frequently Used Discrete Distributions

(4 cards)

Uniform distribution on $$N$$ points

Front

$$p(x) = \frac{1}{N}$$ for $$x = 1, 2, ..., N$$

$$E[X] = \frac{N+1}{2}$$

$$Var[X] = \frac{N^2-1}{12}$$

Back

Poisson distribution with parameter λ

Front

Often used as a model for counting the number of events of a certain type that occur in a certain period of time (suppose $$X$$ represents the number of customers arriving for service at a bank in a one hour period, with a mean of λ - the number arriving for service in two hours would have a Poisson distribution with parameter 2λ)

$$p(x) = \frac{e^{-λ}λ^x}{x!}$$ for $$x = 0, 1, 2, ...$$

$$E[X] = Var[X] = λ$$

Back

Binomial distribution with parameters $$n$$ and $$p$$

Front

A single trial of an experiment results in success with probability $$p$$ or failure with probability $$1-p=q$$. If $$n$$ independent trials of the experiment are performed, and $$X$$ is the number of successes that occur, $$X$$ is said to have binomial distribution denoted $$X~B(n,p)$$.

$$p(x) = \binom{n}{x}p^x (1-p)^{n-x}$$ for $$x = 0, 1, 2, ..., n$$, the probability of exactly $$x$$ successes in $$n$$ trials.

$$E[X] = np$$

$$Var[X] = np(1-p)$$

Back

Geometric distribution with parameter $$p$$

Front

$$X$$ represents the number of failures until the first success of an experiment with probability $$p$$ of success.

$$p(x) = (1-p)^x p$$ for $$x = 0, 1, 2, ..., N$$

$$E[X] = \frac{1-p}{p}$$

$$Var[X] = \frac{1-p}{p^2}$$

Back

Frequently Used Continuous Distributions

(11 cards)

Link between exponential and Poisson distribution

Front

Let $$X$$ represent the time between successive occurrences of some type of event, where $$X$$ has an exponential distribution with mean $$\frac{1}{λ}$$ and time is measured in some appropriate units (seconds, minutes, hours, days, etc.)

Let $$N$$ represent the number of events that have occurred when one unit of time has elapsed. Then $$N$$ will be a random variable that has a Poisson distribution with mean $$λ$$.

Back

Integer correction for normal approximation of discrete random variables

Front

If $$X$$ is discrete and integer-valued, then integer correction may be applied in the following way:

$$P[n < X < m]$$ is approximated using a normal variable $$Y$$ with the same mean and variance as $$X$$ and finding probability $$P[n -\frac{1}{2} ≤ Y ≤ m +\frac{1}{2}]$$

Back

Minimum of a collection of exponential random variables

Front

If independent random variables $$Y_{1}$$, $$Y_{2}$$, ..., $$Y_{n}$$ have exponential distributions with means $$\frac{1}{λ_{1}}$$, $$\frac{1}{λ_{2}}$$, ..., $$\frac{1}{λ_{n}}$$, $$Y = min$${$$Y_{1}, Y_{2}, ..., Y_{n}$$} has an exponential distribution with mean $$\frac{1}{λ_{1}+λ_{2}+...+λ_{n}}$$

Back

Normal Distribution

Front

Normal distribution $$X$$~$$N(μ,σ^2)$$ has a mean of $$μ$$ and variance of $$σ^2$$

$$f(x) = \frac{1}{σ\cdot \sqrt{2π}}\cdot e^{-\frac{(x-μ)^2}{2σ^2}}$$

$$E[X] =μ$$

$$Var[X] = σ^2$$

Back

Standardizing normal random variables

Front

For normal random variable $$X$$~ $$N(μ,σ^2)$$, find $$P[r < X < s]$$ by standardizing $$Z = \frac{X - μ}{σ}$$. Then

$$P[r < X < x] = P[\frac{r-μ}{σ} < \frac{X-μ}{σ} < \frac{s-μ}{σ}] = ɸ(\frac{s-μ}{σ}) - ɸ(\frac{r-μ}{σ})$$

Back

Combination of normal random variables $$W = X_{1} + X_{2}$$

Front

$$W$$ is also a normal random variable with mean $$μ_{1} + μ_{2}$$ and variance $$σ_{1}^2 + σ_{2}^2$$.

Back

Exponential distribution with mean $$\frac{1}{λ} > 0$$

Front

Typically used to model the amount of time until a specific event occurs.

$$f(x) = λe^{-λx}$$ for $$x > 0$$

$$F(x) = 1 - e^{-λx}$$

$$S(x) = e^{-λx}$$

$$E[X] = \frac{1}{λ}$$

$$Var[X] = \frac{1}{λ^2}$$

Lack of memory property: $$P[X > x + y|X > x] = P[X > y]$$

Back

Uniform distribution on interval (a,b)

Front

$$f(x) = \frac{1}{b-a}$$

$$F(x) = \int_{a}^{x} f(x) dx = \frac{x-a}{b-a} a ≤ x ≤ b$$

$$E[X] = \frac{a+b}{2}$$

$$Var[X] = \frac{(b-a)^2}{12}$$

Back

Finding 95th percentile of normal variable $$X$$~$$N(1,4)$$

Front

$$P[X ≤ c] = .95$$

$$P[\frac{X - 1}{\sqrt{4}} ≤ \frac{c - 1}{\sqrt{4}}] = ɸ(\frac{c - 1}{\sqrt{4}}) = .95$$

$$\frac{c - 1}{\sqrt{4}} = 1.645$$

$$c = 4.29$$

Back

Standard Normal Distribution

Front

Standard normal distribution $$Z~N(0,1)$$ has a mean of 0 and variance of 1. A table of probabilities is provided on the exam.

$$f(x) = \frac{1}{\sqrt{2π}}\cdot e^{-\frac{x^2}{2}}$$

$$E[X] = 0$$

$$Var[X] = 1$$

Back

Gamma distribution with parameters $$n > 0$$ and $$β > 0$$

Front

$$f(x) = \frac{β^n \cdot x^{n-1} \cdot e^{-βx}}{(n-1)!}$$

$$E[X] = \frac{n}{β}$$

$$Var[X] = \frac{n}{β^2}$$

Back

Joint, Marginal, and Conditional Distributions

(11 cards)

Independence of random variables $$X$$ and $$Y$$

Front

$$X$$ and $$Y$$ are independent if the probability space is rectangular (endpoints can be infinite) and

$$f(x,y) = f_{X}(x)\cdot f_{Y}(y)$$

Which is equivalent to

$$F(x,y) = F_{X}(x)\cdot F_{Y}(y)$$ for all $$x,y$$

Back

$$E[E[X|Y]] =$$

Front

$$=E[X]$$

Back

Marginal distribution of $$X$$ found from a joint distribution of $$X$$ and $$Y$$

Front

If $$X$$ and $$Y$$ have a joint distribution with density function $$f(x,y)$$, the marginal distribution of  $$X$$ has a density function $$f_{X}(x)$$ which is equal to

$$f_{X}(x) = \sum_{y} f(x,y)$$ in the discrete case and

$$f_{X}(x) = \int f(x,y)dy$$ in the continuous case.

Note that $$F_{X}(x) = \lim_{y \to \infty} F(x,y)$$

Back

Covariance between random variables $$X$$ and $$Y$$

Front

$$Cov[X,Y] = E[XY] - E[X]E[Y]$$

Also, note

$$Var[aX+bY+c] = a^2 Var[X] + b^2 Var[Y] + 2abCov[X,Y]$$

Back

Expectation of a function of jointly distributed random variables

Front

Discrete case:

$$E[h(X,Y)] = \sum_{x}\sum_{y} h(x,y)\cdot f(x,y)$$

Continuous case:

$$E[h(X,Y)] = \int \int h(x,y)\cdot f(x,y)dydx$$

Back

Joint distribution of variables $$X$$ and $$Y$$

Front

The probability $$P[(X = x)\cup (Y = y)]$$ for each pair $$(x,y)$$ of possible outcomes.

For discrete random variables:

(i) $$0 ≤ f(x,y) ≤1$$ and

(ii) $$\sum_{x}\sum_{y} f(x,y) = 1$$.

For continuous random variables:

(i) $$f(x,y) ≥ 0$$ and

(ii) $$\int_{-\infty}^{\infty} f(x,y)dydx = 1$$

Back

$$Var[X]$$ using conditional distributions

Front

$$Var[X] = E[Var[X|Y]] + Var[E[X|Y]]$$

Back

Coefficient of correlation between random variables $$X$$ and $$Y$$

Front

$$\frac{Cov[X,Y]}{σ_{X}σ_{Y}}$$

Back

Moment generating function of a joint distribution

Front

$$M_{X,Y}(t_{1}, t_{2}) = E[e^{t_{1}x + t_{2}y}]$$

$$E[X^n Y^m] = \frac{∂^{n+m}}{∂^n t_{1}∂^m t_{2}} M_{X,Y}(t_{1}, t_{2})$$ evaluated at $$t_{1} = t_{2} = 0$$

Back

Pdf of a uniform joint distribution of $$X$$ and $$Y$$ on region $$R$$

Front

$$f(x) = \frac{1}{Area of R}$$

Probability of event A $$P[A] = \frac{Area of A}{Area of R}$$

$$f_{Y|X}(Y|X=x)$$ has a uniform distribution on the line segment defined by the intersection of the region $$R$$ with the line $$X=x$$

Back

Conditional distribution of $$Y$$ given $$X = x$$

Front

$$f_{Y|X}(y|X = x) = \frac{f(x,y)}{f_{X}(x)}$$

$$E[Y|X = x] = \int y\cdot f_{Y|X}(y|X = x)dy$$

If $$X$$ and $$Y$$ are independent, then

$$f_{Y|X}(y|X = x) = f_{Y}(y)$$

Also note that

$$f(x,y) = f_{Y|X}(y|X = x)\cdot f_{X}(x)$$

Back

Important Formulas to Memorize

(5 cards)

Infinite sum of a geometric progression $$a + ar + ar^2 + ...$$

Front

$$a + ar + ar^2 + ... = \frac{a}{1-r}$$

Back

$$\int xe^{ax} dx =$$

Front

$$\int xe^{ax} dx = \frac{xe^{ax}}{a} - \frac{e^{ax}}{a^2}$$

Back

$$\int_{0}^{\infty} x^ne^{-cx} dx$$

Front

$$\int_{0}^{\infty} x^ne^{-cx} dx = \frac{n!}{c^{n+1}}$$

Back

Sum of the first $$n$$ terms of an arithmetic progression $$a + a + d + a + 2d + a + 3d + ... + a + (n-1)d$$

Front

$$a + a + d + a + 2d + a + 3d + ... + a + (n-1)d = na + d\cdot \frac{(n)(n-1)}{2}$$

Back

Sum of the first $$n$$ terms of a geometric progression $$a + ar + ar^2 + ... + ar^{n-1}$$

Front

$$= a\cdot \frac{1-r^n}{1-r}$$

Back

Functions and Transformations of Random Variables

(11 cards)

Distribution of the sum of $$k$$ independent Normal variables

Front

$$Y$$~$$N(\sum μ_i,\sum σ_i^2)$$

Back

Pdf of $$Y = u(X)$$

Front

Can be found in two ways:

(i) $$f_Y(y) = f_X(v(y))\cdot|v'(y)|$$

(ii) $$F_Y(y) = F_X(v(y)), f_Y(y) = F'_Y(y)$$ for strictly increasing functions

Back

Central Limit Theorem

Front

If $$X$$ is a random variable with mean μ and standard deviation σ and $$Y = X_1 + X_2 + \cdots + X_n$$, then $$E[Y] = nμ$$ and $$Var[Y] = nσ^2$$

As $$n$$ increases, the distribution of $$Y$$ approaches a normal distribution $$N(nμ,nσ^2)$$

If an exam question asks for probability involving a sum of a large number of independent random variables, it is usually asking for the normal approximation to be applied.

Back

Distribution of the sum of $$k$$ independent Poisson variables

Front

$$Y$$~$$P(\sum λ_i)$$

Back

Distribution of the sum of independent discrete random variables

Front

$$P[X_1+X_2 = k] = \sum_{x_1=0}^k f_1(x_1)\cdot f_2(k-x_1)$$

Back

Distribution of the sum of continuous random variables $$Y = X_1 + X_2$$

Front

$$f_Y(y) = \int_{-\infty}^{\infty} f(x_1, y-x_1)dx_1$$

For independent continuous variables,

$$f_Y(y) = \int_{-\infty}^{\infty} f_1(x_1)\cdot f_2(y-x_1)dx_1$$

If $$X_1>0$$ and $$X_2>0$$, then

$$f_Y(y) = \int_{0}^{y} f(x_1, y-x_1)dx_1$$

Back

Distribution of the sum of discrete random variables

Front

$$P[X_1+X_2 = k] = \sum_{x_1=0}^k f(x_1, k-x_1)$$

Back

Transformation of $$X$$

$$Y = u(X)$$

Front

$$u(x)$$ is a one-to-one function, strictly increasing or decreasing, with inverse function $$v$$ so that $$v(u(x)) = x$$

$$Y = u(X)$$ is referred to as a transformation of X.

Back

Moment generating function of a sum of random variables

Front

$$M_Y(t) = M_{x_1}(t)\cdot M_{x_2}(t)\cdots M_{x_n}(t)$$

Back

Distribution of the maximum or minimum of a collection of random variables

Front

$$X_1$$ and $$X_2$$ are independent random variables.

$$U = max{X_1,X_2}$$ and $$V = min{X_1,X_2}$$

$$F_U(u) = P[(X_1≤u)\cap (X_2≤u)] = P[(X_1≤u)]\cdot (X_2≤u)] = F_1(u)\cdot F_2(u)$$

similarly

$$F_V(v) = 1 - [1 - F_1(v)]\cdot [1 - F_2(v)]$$

Back

Distribution of the sum of $$k$$ independent binomial variables

Front

$$Y$$~$$B(\sum n_i ,p)$$

Back

Risk Management Concepts

(2 cards)

Policy Limit

Front

A policy limit of amount u indicates that the insurer will pay a maximum amount of $$u$$ when a loss occurs. The amount paid by the insurer is $$X$$ if $$X≤u$$ and $$u$$ if $$X>u$$.

The expected payment is $$\int_0^u [1-F_X(x)]dx$$

Back

Deductible Insurance

Front

If a loss of amount $$X$$ occurs, the insurer pays nothing if the loss is less than $$d$$ and pays the amount of the loss in excess of $$d$$ if the loss is greater than $$d$$.

$$Y =$$

$$0$$ if $$X≤d$$

$$X-d$$ if $$X>d$$

$$= Max(X-d,0)$$

The expected payment is $$\int_d^{\infty} (x-d)f_X(x)dx = \int_d^{\infty} [1-F_X(x)]dx$$

Back