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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How often does it happen that the oldest person alive dies?

Today, we are brought the sad news that Europe's oldest woman died. A little over a week ago the oldest person in the U.S. unfortunately died. Yesterday, the Netherlands' oldest man died peacefully. ...
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1answer
486 views

Calculating a continuously varying, continuously paid annuity

I'm unsure how to calculate a continuously varying, continuously paid annuity. I'll write up my solution (which I suspect is wrong) to one, sample question, and I would greatly appreciate any ...
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3answers
298 views

Which methods are used by actuaries in practice?

Recently I read a comment from an actuary that a lot of the math they studied as part of the program they never actually used. I'm not interested in becoming an actuary, but I'm interested in ...
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2answers
124 views

Memorylessness and Expectation

I have a specific problem I'm working on. Let $X$ be an exponential random variable, and let $Y$ be a random variable defined by: $$ Y = \begin{cases} 0 & \text{ if } X < d \\ ...
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1answer
86 views

Monthly rental fee to achieve given profit on average, given probabilities of numbers of rentals

I have this problem here and I'm very unsure of how to start this. I have an idea but I'm not sure where to go from a certain point. The problem says: A video rental store is analyzing a flat fee ...
4
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1answer
135 views

A question in Finan's FM/2 book

Problem 6 on page 47 of Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 is: Fund A is invested at an effective annual ...
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3answers
2k views

Masters in Actuarial Science

I am applying to a grad school for the Masters in Actuarial Science. Now i am getting cold feet. I do love math, i was always good in math (not excellent or a genius). Did all adv. calculus classes ...
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4answers
15k views

Are the actuarial exams hard? [closed]

I heard that they are difficult. Is this true? Are they like the qualifying exams in grad school? For example, is the probability exam and the financial math exam comparable to qualifying exams (e.g. ...
3
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2answers
46 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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3answers
75 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
87 views

SOA Exam P Question: $P$ is a random point on the Cartesian Coordinate Plane. Find the variance of the area of a circle formed by $P$.

Caution: This problem was "passed down" to me and I think the wording was altered or lost along the way. I will post the problem as I have it and then make suggestions on what I think it should be. ...
3
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1answer
1k views

Confused about Effective Rate of Discount- Theory of Interest

I'm currently reading Kellison's book, The Theory of Interest. I've reached the chapter on Effective Rate of Discount and it's somewhat confusing. The book explains it as a loan where interest is paid ...
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3answers
179 views

Expected value and life

Let $e_{x} = \int_{0}^{\infty} p_{x}(t) \ dt$ where $p_{x}(t)$ is the probability that a person aged $x$ will survive at least $t$ more years. Why is $e_{x} \leq e_{x+1}+1$? We know that $e_{x} \geq ...
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3answers
32 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 ...
2
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1answer
80 views

How is this the right answer?

In a survey of customer satisfaction, participants are asked to give a score of 1,2,3 or 4 to each of the 6 questions. If participants are instructed not to give the same numerical score to more than ...
2
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1answer
174 views

Bonds and Force of Interest

Studying for FM/2 and ran into this problem dealing with bonds; A 1,000 par value 3 year bond with annual coupons of 50 for the first year, 70 for the second year, and 90 for the third year is bought ...
2
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2answers
79 views

How would I determine whether these events are independent?

I'm studying for CAS/SOA Exam 1/P and I'm stumped on this question. It says: From the set of families with two children a family is selected at random. Let $X_1=1$ if the first child of the family ...
2
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1answer
67 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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1answer
69 views

Variable substitution in probability

In modeling the number of claims filed by an individual under an automobile policy during a three-year period, an actuary makes the simplifying assumption that for all integers $n \ge 0$, ...
2
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1answer
131 views

SOA Exam P Question: Exponential Distribution

Here is an Exam P problem as I have it. That is, it was passed down to me from someone else and I am unsure if the wording is exactly as it was originally posted. I've tried searching for this ...
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0answers
38 views

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
209 views

Two questions on nominal rates of interest

I'm reading Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 and have difficulty with two of his questions. Problem 9.6 ...
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0answers
112 views

Finding ratio of interest rates

I'm reading through Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 on my own and am unsure how to proceed with a question ...
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1answer
98 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
581 views

What does it mean that the probability density function is proportional to a function?

I'm studying for SOA/CAS Exam P and I have a problem that says that $X$ is a continuous and positive random variable whose probability density function is proportional to: $$\frac{1}{(1+x)^5}$$ Where ...
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2answers
123 views

Solving i for annuities equation without financial calculator

I would like to know if there was a way to approximate i here without a financial calculator, in the following equation: $\displaystyle -50000 + \frac{12992}{1+i} + \frac{12992}{(1+i)^2} + ⋯ + ...
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1answer
34 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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2answers
41 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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1answer
55 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
99 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 ...
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2answers
116 views

Are waiting times always going to be exponentially distributed?

I'm studying for CAS/SOA Exam P/1 and a question I have here is: We have a portfolio of $20$ insurance policies. The number of claims per policy in a $3$-month period has a Poisson distribution ...
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1answer
41 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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1answer
98 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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2answers
235 views

Closed form for Exponential Conditional Expected Value & Variance

I am wondering if there is a closed form for finding the expected value or variance for a conditional exponential distribution. For example: $$ E(X|x > a) $$ where X is exponential with mean ...
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2answers
202 views

Probability of being within one standard deviation of mean given only a moment generating function

I am trying to figure out how to approach this: For starters, I calculated M'(0) and M"(0), which allowed me to find the mean and variance. I got 2 for the mean and 1.6 for the variance. But now, ...
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1answer
470 views

How to find limits of integration on a convolution of CRVs

In finding the convolution of two independent and continuous random variables, I am struggling with limits of integration. I cannot seem to figure out over what intervals the probability density ...
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1answer
203 views

Stuck on solving for x in exponential to find variance

The problem seems simple: Let X be an exponential random variable such that $P(X \le 2) = 2P(X > 4)$. Find the variance of X. Easy, right? $ P(x \le 2) = 1 - e^{-2\lambda} $ and $ P(x > 4) = ...
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2answers
139 views

Where is the logic with this trinomial probability?

I am studying for Exam P to take in a few hours so I'd appreciate very much if someone got to me befoe I left. The problem says that we have three categories of risk, high, medium and low risk. The ...
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1answer
439 views

Converting an Annuity due to Annuity immediate

I'm working on the following problem at the moment while preparing for an exam. Find the present value of payments of 200 every six months starting immediately and continuing through four years from ...
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1answer
311 views

Help understanding confusing present value question (Theory of Interest)

I'm currently enrolled in a theory of interest class that covers the same material that's on exam 2/FM. At the instructors recommendation, I'm working through all of the problems in the book on my own ...
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1answer
59 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
72 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
19 views

Writing and expression to find the probability

Suppose (x) and (y) are independent lives. $T(x)$ and $T(y)$ denotes the future lifetimes. How do I write an expression to find out the probability that exactly one of the dies within the next 10 ...
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1answer
50 views

Probability of $X^{2} > Y^{3}$ over distribution other than uniform

I was working on a problem in which $X$ and $Y$ are continuous random variables which both have the uniform distribution $U[0, 1]$. The question was: what is the probability that $X^2 > Y^3$. ...
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2answers
343 views

Finding moment generating function from piecewise CDF

I am stuck on this problem: I am given the CDF, and unless there is a shortcut I am not remembering, I need to find the PDF before I can get the moment generating function and solve the problem. ...
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1answer
985 views

Increasing and decreasing annuity

Going in circles..... Find an expression for the present value of an annuity-immediate where payments start at 1, increase by 1 each period up to a payment of $n$, and then decrease by 1 each period ...
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1answer
117 views

Continuous annuity calculation

This is a problem from Marcel Finan's Exam FM/2 course. It is not homework but I am studying for the FM exam and trying to get through this. You are given $\frac{d}{dt}\bar{s}_t$ = $(1.02)^{2t}$. ...
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1answer
126 views

$e^{F(x,y)}$ Type Multi-variable Exponential Integrals

I am sure all you integration buffs can do this faster than I can type it. Your help with a quick explanation and solution is appreciated. $$F _{XY} = \int_0^\infty\int_0^\infty xye^{-\frac{x^2 + ...
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1answer
478 views

Procedure for Max Function with identical random variables

I am studying for the P/1 actuarial exam, and I keep encountering one type of problem which I can't seem to solve and can't seem to find any generalized explanation for how to do it in my textbooks. ...
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1answer
84 views

Finding principal on two loans of equal term given different rates and amounts

I'm reading through Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 on my own and am unsure how to proceed with a question ...