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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Deriving the Pareto Distribution from an Exponential Distribution [on hold]

Let $T$ be an exponential random variable with hazard rate $a>1$. Consider a random variable defined by the condition $X = b (e^t - 1)$." I need to find the density of $x$.The answer is ...
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Saving account & annual effective rate

A bank pays 6% per annum on its savings accounts. At the end of every 3 years a 2% bonus is paid on the balance. Determine the annual effective rate of interest, earned by an investor if the deposit ...
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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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53 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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32 views

Calculating Interest

Question: Calculate the amount of interest earned between time $t=3$ and $t=8$ if $i(n)=n$ I was thinking to do like this $3+4+5+6+7+8=33$ but the answer is $30$.
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Confused about Effective Rate of Discount and Effective rate of Interest - Theory of Interst

Question: If $A(0)=500$ and $A(1)=525$, find $i_1$ and $d_1$. From what I understand the equation for $i_n$ is $i_n = \frac{a(n) - a(n-1)}{a(n-1)}$ and the equation for $d_n$ is $d_n = \frac{a(n) - ...
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1answer
68 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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13 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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1answer
122 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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1answer
50 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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33 views

SBC score clarification

For the SBC test, it says that this is more favourable as compared to the Anderson darling, kolomogorov Smirnoff test and the chi-squared tests because it doesn't have the fault that, when n increases ...
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1answer
44 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
24 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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2answers
52 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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1answer
12 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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40 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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52 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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45 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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39 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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1answer
33 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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56 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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83 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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10 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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33 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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3answers
43 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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33 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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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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70 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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27 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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63 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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68 views

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
26 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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“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
33 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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29 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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86 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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1answer
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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1answer
54 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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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
62 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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3answers
41 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 ...
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Expected value problem with given condition.

the following is the problem I am working on. A fair die is rolled repeatedly. X is the necessary number of rolls until 5 shows up, and Y is the necessary number of rolls until 6 shows up. Find ...
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1answer
104 views

Soft question regarding the P-exam for actuaries.

I am planning to take the P-exam soon and have been studying and practicing some probability problems. And for those who have taken the test, I have a question. I have noticed that some of the ...
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3answers
35 views

Deriving a property regarding variance.

I am studying for the P-exam for actuaries and I've encountered a property that said, $Var(x\pm y)=Var(x)+Var(y)$ I come from a math major and it has been years since I was taught statistics or ...
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1answer
52 views

nominal rates and effective rates

I would like some help understanding some basic concepts about converting nominal rates into effective rates, and vice-versa. Some of the terms are a little confusing to me. Some examples I would ...
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1answer
102 views

Interest Rate Tree in Matlab

I would like to calibrate a interest rate tree using the optimization tool in matlab. Need some guidance on doing it. The interest rate tree looks like this: How it works: 3.73% = 2.5%*exp(2*0.2) ...
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2answers
50 views

Probability question : meaning of the sentence

The following is the problem I am working on. The probability of a passing car being an import is defined as $p(i)=1/4$ and the probability of it being domestic is $p(d)=3/4$. Find the probability ...
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2answers
24 views

pdf vs cdf -random variables-

The following problem is what I am working on. $F(x)=\frac{1}{1+e^{-x}}$ is the cumulative density function defined for all real numbers. Find the probability density function. My understanding ...