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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Question regarding interest method during a year. $i-\delta$

B and P deposits the amount $100$ in separate bank accounts. B's account have a nominal rate convertible semiannually. P's account has a force of interest $\delta$. After $7.25$ years, each ...
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63 views

Force of interest problem from Exam FM

Tawny makes a deposit into a bank account which credits interest at a nominal interest rate of $10\%$ per annum, convertible semiannually. At the same time, Fabio deposits $1000$ into a different ...
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An actuary question but more of an algebraic manipulation

I'm stuck in this actuary question. If $a_{\overline{n|}}=x$ and $a_{\overline{2n|}}=y$, express $d$ as a function of $x$ and $y$ Hints: $a_{\overline{n|}}=\frac{1-v^n}{i} $, $v=\frac{1}{1+i}$, ...
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Force of interest :$e^{\int_0^t{\delta_t}ds}$ vs $(1+\frac{i^{(2)}}{2})^2$

I was wondering if someone could help me confirm the answer for the following problem regarding force of interest. X deposits $1$ at time $t=0$ with force of interest $\delta_t=\frac{t^2}{k}$. Y ...
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69 views

182-Day T-Bill vs. 91-Day T-Bill

I am trying to understand how T-Bills work and it would be great if someone could explain me using the following question At $t=0$ Smith buys a 182-Day T-Bill with a simple annual discount rate of ...
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64 views

Discount rates vs. Interest rate problem.

I am working on a problem as follows. A discounted value $X$ that is due when $t=0.5$ has a present value \$4992. Calculate the value of $X$ when a), the effective annual compound interest rate ...
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28 views

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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36 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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22 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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37 views

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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19 views

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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171 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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2answers
75 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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2answers
46 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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18 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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113 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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16 views

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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118 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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85 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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26 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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19 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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343 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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65 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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49 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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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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96 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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29 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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80 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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80 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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55 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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51 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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131 views

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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52 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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93 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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2answers
104 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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18 views

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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37 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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52 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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47 views

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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36 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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4answers
127 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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41 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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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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35 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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72 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
47 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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230 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 ...