Actuarial science is a discipline that uses mathematics and statistics to assess risk. The mathematics involved in actuarial science includes probability, statistics, finance, life insurance mathematics, and more.

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Finance problem - Canadian bond with simple interest

I am having trouble understanding a lot of things from the following problem. It would be very helpful if I could get some explanations. Smith purchases a Canadian bond for 1000 with an issue ...
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Financial math vocab; “convertible”, “roll over”?

I am having trouble understanding what is going on regarding the following problem. Smith receives income from his investments in yen. He finds a bank that will issue a term deposit that allows ...
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34 views

Am I plugging this into the calculator correctly?

I want to make sure I am plugging these into my BA II plus correctly in order to help me when I'm taking an exam. I keep looking up how to work the calculator and I think I have gotten the steps down ...
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21 views

Problem regarding Investments and selling the right.

I have trouble solving the following problem algebraically. Smith lends $\$1000$ to Jones at time $t=0$. Jones is supposed to repay Smith by paying $\$100$ at time $t=1$ and $2$, and $\$1000$ at ...
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Proof $(1+i)^t < 1+it$ if $0 < t < 1$

I am trying to prove that compound interests grow slower than simple interest in the first year. I attempted to do this with the following approach. Let $$f(t) = (1+i)^t$$ then $$f'(t) = ...
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Find expression for $\frac{d}{dn}\overline{s}_{\overline{n|}}$

I got a question from book Mathematics of Investment and Credit question 2.3.25 b) Find expression for: $$\frac{d}{dn}\overline{s}_{\overline{n|}}$$ I got the solution, but there is one part that I ...
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24 views

How to interpret this annuity-immediate equation?

I am trying to verbally interpret this annuity equation of annuity immediate, payable m-thly $$a^{(m)}_{\overline{n|}i}=\frac{i}{i^{(m)}}a_{\overline{n|}}$$ I drew a time line and try to make ...
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167 views

Expressing a summation using matrix algebra

Consider the $n \times r$ matrix $$\begin{pmatrix} X_{11} & X_{12} & \cdots & X_{1n} \\ X_{21} & X_{22} & \cdots & X_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ...
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70 views

Are there other accumulation functions that holds $a(n-t)={a(n) \over a(t)}$?

This might be a beginner's question regarding accumulation methods and their functions, but so far I have learned that compound interest satisfy $$a(n-t)={a(n) \over a(t)}$$ Which allows nice ...
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45 views

What is a discount?

I am learning some financial terms and am having trouble understanding what a discount $d$ is. Numerically, I understand that it is defined as $\frac{i}{1+i}$ but I do not intuitively understand what ...
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16 views

Different types of interest

I'm currently taking actuarial mathematics one. I have a quiz and exam coming up, but I'm very confused between all these interest rates and what they represent. There is annual effective interest ...
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110 views

conditional expected value - Poisson process plus random variable

I've struggled with this actuary excercise for a while and I don't know how to do it: Each claim can be characterized by two random variables $(T,D)$, where $T$ is the moment of reporting the claim ...
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Expected payment problem. $d=4, u=5, f(x)=.125e^{-x/8}$

I am working on the following problem. Let $X$ be a random variable for a loss that is exponential with mean 8. Let $Y$ be the payment by the insurance company given a positive loss, with ...
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2answers
99 views

Pure Mathematics to Actuarial Science.

I did my undergraduate in pure mathematics and recently after a three year break I decided to get a master's degree in Actuarial Science. I am in the second week of classes and I feel completely lost. ...
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1answer
79 views

Fail time with joint density $f_{X,Y}(x,y)=6e^{-x}e^{-2y}, \quad 0<x<y< \infty$

The following is the problem I am solving for. Let $X$ and $Y$ be the fail time for a machine with two components. The component $Y$ will start working if and only if component $X$ fails. The ...
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25 views

Force of mortality calculation

The force of mortality of a non smoker and smoker aged x is μx and c.μx respectively (c>1). What is the probability that a non smoker live longer than a smoker? I guess I haven't really understand ...
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1answer
18 views

Let $f(x,y)=2(x+y)$ for $-<x<y<1$ Find $E[Y]$

This may be a dumb question, but I thought that to get the expected value, we were supposed to calculate $$A = \int_0^1 \int_x^1 y 2(x+y) dydx$$ which gave me a different answer to $$B = \int_0^1 ...
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265 views

Darth Vader Rule: what is the reason for its name, and a formal proof?

I often hear the term "Darth Vader Rule" when calculating the expected value using the survival function and taking the integral where it is defined. I am not quite sure why it is called that (is it ...
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63 views

Unfamiliar notation. (actuarial science)

I am a math instructor self studying for the actuarial exam and I am trying to understand the following notation that I have encountered today. $$E[X \land d]$$ The explanation in the book told me ...
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47 views

Finding the variance problem

I am working on the following problem and the explanation was not clear to me, so I am seeking for help. The following is the problem. A fire occurs with a probability of 0.01. The damage Y ...
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1answer
26 views

128 factories close down first

I found this problem on http://poissonlabs.com/blog/how-insurance-works. I'm not quite sure how to solve it. Suppose that $1000$ factories belong to an industry, and that each factory faces an ...
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85 views

Maximum of Three Uniform Random Variables

Here is the question, I am studying for exam P, and am using a study guide with a solution guide. I am stumped on this problem, and the solution in the back was very confusing. Any clarity I can get ...
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25 views

Deffered annuity with perpetuity

An annuity immediate has $40$ initial quarterly payments of $20$ followed by perpetuity of quarterly payments of $25$ starting in the eleventh year. Find the present value at $4\% $ convertible ...
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69 views

Interest Theory- Annuity Withdrawals/Deposits

"Consider an investment of $5,000 at 6% convertible semiannually. How much can be withdrawn each half−year to use up the fund exactly at the end of 20 years?" To solve this problem, an equation of ...
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71 views

When to use alternate parametrization of Gamma distribution?

In Loss Models, 4th ed., by Klugman et al., the following parametrization is given for the Gamma distribution: $$f(x) = \dfrac{(x/\theta)^{\alpha}e^{-x/\theta}}{x\Gamma(\alpha)}\text{.} $$ When ...
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51 views

Proof of $\text{TVaR}_p(X)$ and $\text{VaR}_{u}(X)$ relationship

From Loss Models, 4th ed., by Klugman et al.: Definition. Let $X$ denote a loss random variable. The Value-at-Risk of $X$ at the $100p$% level, denoted $\text{VaR}_{p}(X)$ or $\pi_p$, is the ...
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49 views

Actuarial Science - Amortization

Kevin takes out a $10$-year loan of $L$, which he pays by the amortization method at an annual effective interest rate of $i$. Kevin makes payments of $1000$ at the end of each year. The total amount ...
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Prove that $\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}$ is a strictly decreasing function.

This is part of an actuarial science problem. Unfortunately, the official solution of this problem takes the derivative of $$\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}\text{, } \quad x \geq 0\text{.}$$ and ...
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46 views

Two definitions of the Incomplete Gamma Function - are they equivalent?

From Loss Models, 4th ed., by Klugman et al.: Definition 5.5 The incomplete Gamma function with parameter $\alpha > 0$ is denoted and defined by $$\Gamma\left(\alpha ; x\right) = ...
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78 views

Beta distribution for exam P?

I had a quick question regarding the beta distribution and exam P for actuaries. From the recommended books that I have seem, beta distribution does not seem like it is likely to show up on the P ...
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97 views

How to remember the Jacobian

the following is the problem that I was working on. Let $f(x,y)=8xy$ for $0<x<y<1$. What is the joint density function of $W={X \over Y}$ and $Z=Y$? Since I am self studying this ...
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The average of random samples problem

the following is the problem that I am working on. A random sample of size 16 is to be taken from a normal population having mean 100 and variance 4. What is the 90th percentile of the ...
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34 views

Finding the correlation coefficient of ordered statistics

I am working on the following problem. Let $$X_{(1)}, \ldots ,X_{(n)}$$ be the order statistics from the uniform distribution of $[0,1]$. Find the coefficient correlation of $X_{(1)}$ and ...
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51 views

Convergence of the series $\sum\limits_{n=1}^{\infty}nkr^{n-1}$, $|r| < 1$

One series that comes up in actuarial science courses is $$\sum\limits_{n=1}^{\infty}nkr^{n-1},\quad |r| < 1\text{, }k>0\text{.}$$ [Typically $|r| < 1$, but I'm wondering if this is a ...
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Definition of Percentile (undefined notation)?

From Loss Models, 4th ed., by Klugman et al.: Definition 3.6 The $100p^{\text{th}}$ percentile of a random variable is any value $\pi_p$ such that $F\left(\pi_{p^{-}}\right) \leq p \leq ...
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33 views

Find the density function of $2T_1+T_2$

A device containing two key components fails when, and only when, both components fail. The lifetimes, $T_1$ and $T_2$, of these components are independent with common density function $f(t) = ...
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117 views

Remembering the mean and variance of Poisson vs Exponential

I am having the P-Exam for actuaries on September and I am trying to work on my details. One of the troubles that I have is to remember the difference between Poisson and Exponential distribution. I ...
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38 views

Which model to use ? (probability problem)

The following was the problem that I was working on. As a part of the underwriting process for insurance, each prospective policyholder is tested for high blood pressure. Let X represent the ...
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1answer
74 views

What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?
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Pdf of $Z=(XY)^{1/2}$. with X,Y independent r.v. with the same distribution (iid) [closed]

Let be $X,Y$ two independent random variables having the same distribution (the following is the density of this distribution) $$f(t)= \frac{1}{t^2} \,\,\, \text{for $t>1$}$$ Calculate the ...
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1answer
33 views

Conditional probability problem with three events

I am solving this problem. Given... $P[W|T]=0.8, \space P[W|T \cap G]=.65, \space P[W | G' \cap T]=1$ Find $P[G'|T]$ I understand that what I am looking for is $$P[G'\cap T]/P[T]$$ or ...
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70 views

“Taxes and Option Prices” (question about Derivatives Markets by McDonald)

Thanks in advance for any help, and please tell me if there's anything I can do to make things clearer. I am having trouble understanding appendix 10.A to Derivatives Markets by Robert L. McDonald. ...
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1answer
46 views

Notation in Actuarial text

I am not very knowledgeable in Actuarial Sciences, but I was looking at a text, and found a notation, which I cannot find any references to. What does: $_{2|2}q_{1} = 0.288$ mean? I don't think it ...
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30 views

Quick question regarding second moment

The following question is what I was working on. a% of the population has a risk of incurring damage that has a Poisson distribution with mean 1. Similarly, b% has a distribution with mean 2 and ...
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206 views

Difference between conditional and intersection in probability.

I am having hard time figuring out if it is a conditional probability or an "and" probability under the following types of problems. When a student is absent, the probability of the student being ...
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38 views

Insurance problem

The following is the type of problem that I am dealing with. An insurer makes $n=4$ driving errors, each independently resulting in an accident with probability $p=.3$. Each accident results in a ...
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64 views

Actuarial science problem with cdf

The following is the problem that I am working on. A loss random variable X has the following cdf: $$F(x)= 0 {\space} \text{if x<0}, .2+.3x {\space} \text{if $0 \le x<2$}, {\space}1 ...
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97 views

Probability of remaining lifetime using force of mortality

I've been stuck on this question for the past half hour and still have no idea how to solve it... I don't think it's supposed to be very difficult but I'm struggling: There are two independent live ...
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2answers
74 views

Joint density functions with $e$

This is the last question in the joint density function section of the packet I'm using to study for the actuarial exams and I'm intimidated by the question. I'm sure it's not overly difficult, I'm ...
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49 views

Bayesian Statistics

Question: Given $N$, $X$ is distributed as $\mathrm{B}(N,\theta)$. Derive the unconditional distribution of X assuming N is distributed as $P(\lambda)$. This is what I have tried so far: $$x|N \sim ...