0
votes
1answer
29 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 ...
0
votes
1answer
22 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 ...
1
vote
2answers
42 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 ...
1
vote
2answers
36 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 ...
1
vote
1answer
41 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 ...
0
votes
2answers
28 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 ...
0
votes
1answer
30 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) = ...
2
votes
4answers
63 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 ...
1
vote
1answer
26 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 ...
4
votes
1answer
58 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 $\exp(╬╗)$, what is the pdf of $Z=X/\max(X,Y)$?
0
votes
2answers
54 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 ...
1
vote
1answer
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 ...
0
votes
0answers
51 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 ...
0
votes
1answer
36 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 ...
1
vote
1answer
52 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 ...
1
vote
0answers
66 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 ...
1
vote
2answers
61 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 ...
0
votes
0answers
29 views

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 ...
2
votes
3answers
34 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 ...
3
votes
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 ...
3
votes
3answers
90 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 ...
0
votes
1answer
37 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
1
vote
1answer
195 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 ...
0
votes
1answer
57 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 ...
1
vote
1answer
66 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 ...
0
votes
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 ...
1
vote
1answer
109 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.
2
votes
1answer
96 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 $n \ge 0$, ...
4
votes
2answers
140 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 \\ ...
1
vote
0answers
119 views

How to derive the variance premium formula

How to derive formula (5.5) with Taylor's expansion in the following link? The difficult part of my question is that i had never seen Var(S) appear in Taylor's expansion ...
1
vote
1answer
20 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 ...
3
votes
1answer
96 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. ...
0
votes
2answers
69 views

What is a two point support in this lemma?

What is the terminology of two point support in this lemma?
0
votes
1answer
71 views

How to calculate $\Pr[\max(X,Y)<4]$?

Suppose the joint PDF of X,Y is $f(x,y)=1/40$ and $0 < x < 5$ and $0 < y < 8$. How to calculate $\Pr[\max(X,Y)<4]$?
1
vote
2answers
269 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 ...
1
vote
1answer
53 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 $X^2 > Y^3$. ...
1
vote
2answers
433 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. ...
1
vote
2answers
217 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, ...
2
votes
1answer
561 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 ...
1
vote
1answer
140 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 + ...
1
vote
1answer
228 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) = ...
1
vote
1answer
540 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. ...
1
vote
2answers
168 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 ...
4
votes
1answer
91 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 ...
1
vote
2answers
130 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 ...
1
vote
1answer
684 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 ...
0
votes
3answers
369 views

SOA/CAS Exam P Question (from previous exam (Nov. '09)): Finding percentiles

So this question comes from SOA/CAS Exam P of November 2009, I'm not sure why the solution is the way it is. The question says this: An insurance company sells an auto insurance policy that ...
2
votes
2answers
82 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
votes
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
votes
3answers
180 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 ...