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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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
87 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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48 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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1answer
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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4answers
87 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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35 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
68 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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76 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
27 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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69 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
39 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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1answer
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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0answers
149 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
55 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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0answers
88 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
67 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
115 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
36 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
62 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
106 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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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
25 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 ...
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1answer
129 views

Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
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102 views

Probability problem - fake and real diamonds -

A box contains 35 gems, of which 10 are real, 25 are fake. Gems are randomly taken out of the box, one at a time without replacement. What is the probability that exactly 2 fakes are selected ...
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1answer
50 views

poisson and discrete distribution

Business failures are due to three mutually exclusive risks: market risk, credit risk, and operation risk, which account for 20%, 30%, and 50%, respectively, of all business failures. Suppose the ...
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39 views

variance unchanged under subtracting mean - application in portfolio theory

How to get to even the first step? How to derive http://i.stack.imgur.com/R3TIk.png with given http://i.stack.imgur.com/3aLAE.png
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1answer
67 views

Questions on share prices of a Company.

Company Z is currently financed solely by common stock and has 1000 outstanding shares with a (time 0) market price of 10 dollars per share. The company’s expected earnings is 1000 dollarseach year ...
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65 views

Discounting Perpetuity Question

"A project pays a dividend of $0.75 next year and then grows at 12% for 3 more years, and then grows at 8% indefinitely thereafter, find PV" Okay so first step is to find the initial value of ...
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1answer
271 views

Probability problem from insurance test

I am renewing my probability knowledge and I am having trouble trying to solve some exercises. An insurance company pays hospital claims. The number of claims that include emergency room or ...
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82 views

Continuous pyments with continuous compounding

At time $t=0$ saving account balance is $0$. Then we start continuous payments with intensivity $C_t$. Continuous compounding intensivity is $\delta_t=\frac{1}{1+t}$. Accumulated value of funds at ...
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Reinvesting the interest (generalized version)

If I deposit \$1 at $t=0$ into an account which credits interest at the end of each year at a force of interest $\delta_t$ (assume it's integrable.) Then, if I reinvest the interest at an annual ...
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1answer
48 views

Question on duration matching and reddinggton's immunisation

An insurance company has liabilities of 6 million due in 8 years’ time and 11 million due in 15 years’ time. The assets consist of two zero-coupon bonds, one paying X in 5 years’ time and the other ...
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1answer
48 views

financial mathematics question

An investor is interested in purchasing shares of ABC company. The company pays annual dividends, and a dividend payment of 1.2 per share has just been made. Future dividends are expected to grow at ...
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26 views

Poisson Process Basic Question

The sum of independent interarrival times for the poisson process is a gamma random variable. in general does the sum of exponentials have to be independent to sum to gamma? also this would produce ...
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58 views

what is the meaning to evaluate the variance of probability destribution for insurance in general?

what is the meaning to evaluate the variance of probability destribution for insurance in general? What does it do(setting the price, estimate cash reserve or else), also does evaluate the variance ...
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1answer
67 views

what is the formal mathematical relationship between the variance and the odds that the gambler who has smaller budget here?

Here is a part of Bob Anderson's answer in my question (Does variance do any good to gambling game makers?) Suppose you had two gamblers who were flipping coins against one another with fair odds ...
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1answer
86 views

When does variance fail to meet its purpose in mathematical statistics? [closed]

It have shown in a lot of both math and statistics book, however, When the books define the variance, it doesn't give much attention to math based theoretical background, i wonder if some formula that ...
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2answers
65 views

annuities - equations of value

Chuck needs to purchase an item in $10$ years. The item costs $ \$200$ today, but its price inflates at $4 \%$ per year. To finance the purchase, Chuck deposits $ \$20$ into an account at the ...
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2answers
430 views

Perpetuity Immediate Present Value Question

A perpetuity-immediate pays $X per year. Brian receives the first n payments, Colleen receives the next n payments, and Jeff receives the remaining payments. Brian's share of the present value of ...
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1answer
98 views

simple interest rate to product of interest rates

Which simple interest rate over six years is closest to being equivalent to the following: an effective rate of discount of 3% for the first year, an effective rate of discount of 6% for the second ...
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1answer
129 views

Present Value Cash Flow Questions - Discounting

At an annual effective interest rate of i, i > 0 all the following are equal: i. the present value of 10,000 at the end of 6 years ii. the sum of the present values of 6000 at the end ...
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110 views

What is some math subject area that could widely apply to acturial science?

What is some math subject area that could widely apply to actuarial science? I know that an actuary mainly deals with stochastic processes (stochastic calculus) / probabilty, statistics, calculus.
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1answer
63 views

Trying to understand an integral algebraically and conceptually $f'(t) = \frac{Ae^t}{(0.02A+e^t)^2}$

It is defined that $f'(t) = \frac{Ae^t}{(0.02A+e^t)^2}$ is the rate of population. It is also given that the population at $t=0$ is given as 6. Our goal is to find the time $t$ where the population ...
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1answer
130 views

I am having trouble proving this ,$\int^{\infty}_0 x^n e^{-cx} dx = \frac{n!}{c^{n+1}}$.

To be more precise, $n$ is a nonnegative integer and c is a positive constant. $\int^{\infty}_0 x^n e^{-cx} dx = \frac{n!}{c^{n+1}}$ supposedly, but I cannot prove it. I've tried integration by ...