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There are two types of claims that are made to an insurance company. Let $N_{i}(t)$ denote the number of type $i$ claims made by time $t$, and suppose that $left{N_...

Question

There are two types of claims that are made to an insurance company. Let $N_{i}(t)$ denote the number of type $i$ claims made by time $t$, and suppose that $left{N_{1}(t), t geqslant 0ight}$ and $left{N_{2}(t), t geqslant 0ight}$ are independent Poisson processes with rates $lambda_{1}=10$ and $lambda_{2}=1 .$ The amounts of successive type 1 claims are independent exponential random variables with mean $$ 1000$ whereas the amounts from type 2 claims are independent exponential random variables

There are two types of claims that are made to an insurance company. Let $N_{i}(t)$ denote the number of type $i$ claims made by time $t$, and suppose that $left{N_{1}(t), t geqslant 0 ight}$ and $left{N_{2}(t), t geqslant 0 ight}$ are independent Poisson processes with rates $lambda_{1}=10$ and $lambda_{2}=1 .$ The amounts of successive type 1 claims are independent exponential random variables with mean $$ 1000$ whereas the amounts from type 2 claims are independent exponential random variables with mean $$ 5000 .$ A claim for $$ 4000$ has just been received; what is the probability it is a type 1 claim?



Answers

Insurance An insurance policy pays for a random loss $X$ subject to a deductible of $C,$ where $0<C<1 .$ The loss amount is modeled as a continuous random variable with density function
$f(x)=\left\{\begin{array}{ll}{2 x} & {\text { for } 0<x<1} \\ {0} & {\text { otherwise }}\end{array}\right.$
Given a random loss $X$ , the probability that the insurance payment is less than 0.5 is equal to 0.64 . Calculate $C .$ Choose one of the following. (Hint: The payment is 0 unless the loss is greater than the deductible, in which case the payment is the loss minus the deductible. Source: Society of Actuaries.
$\begin{array}{llllll}{\text { (a) } 0.1} & {\text { (b) } 0.3} & {\text { (c) } 0.4} & {\text { (d) 0.6 }} & {\text { (e) } 0.8}\end{array}$

Probability Density function is proportional to the one plus extras to minus For now to remove the proportionality sign. We have read Constant K. So we headed here. Okay. No, you're giving in to release Ciro Going finance now, As you know, some of you probably teeth, always won. So integration off the functions would be do one in your case, Constant. So get outside here. No, here we have negative for exponents. No foreign diversity becomes positive. Sorry, minus three. Because negative four plus one is able to minus three. Do you want my industry? No. Three into one is minus city, which is outside or here. Now here we have a minus three sign is in exponents so we can write in denominator with positive sign. Okay, so now it's the UT. A polemic minus lower limit support off us Exit Golden finance miners exited. Siegel. So is we know one buying fine eighties zero and one by one is one. So you two days ago. Negative. So here Casey called three. Now I will tell you off k in organ function. So the hell and three year Our function is therefore faxes. Isn't Goto three into one plus extras too minus four No erected values given by musical do X integration off Extreme do Well thanks. Now you're given intervals Ciro too. And finally, No, we're for fixes given no trees, constant. So take it outside Over here. No Substitute you. Izzy, Go do one plus X here. Now make subject X or exceeds the total U minus one. Now take delivery to off You is equal to one classic. So he becomes Do you delivered to you off One is Siro soh Year zero And, uh, Nativity War exists. Deitz. So he will do you good. D x so we can say d ecstasy. Gordon, do you so here. D X So the taken right here, do you? No, Here X is equal. Do you minus one? Here we have X is Indian Army. Sorry, new editor. So here. See you is 66 Easy. Would you mind this one? I've been here, Rex. Physical toe. You're minus one Indian. A minute here. One minus one is sealed. So let me help you. No separate. Do you know me? Not her. No one minus four years ago to dynasty deal minus four years ago to minus four here within exponents. Now in digression. Off you just rewind the threes. You restroom minus drew upon ministry. And your restroom minus t upon history. No, you're being negative. Sign is in exponents, so Well, did you send a recon righties, denominator? Now put back using one for sex. So your usable to one plus X for here and here. Now execute upper limit minus lower limit. So fast. Excessive vocal in finite for you. I'm here. My no sex. You are zero. Or here and here. No, in finite. Where is Abel doing? Finance in front of you. Busy world Also in finite. Yeah, one squared is equal to one, So he'll be here only minus two more Here and a minus tree into one. Q B's minus three Over here is you know, one buying financially Segal here. I've been here one buying finance. He's also a little but deliver this gate are expected. Really? Lease one by two. So that is our option. C So since he's correct answer. Thank you

Strange, given that probability density function is proportional one plus extras to minus words for your next couple snow, one glass takes. There's truth in this for to remove the proportionality sign. We have had a constant. Okay. Well, as you know, some of the probabilities wants will be. The Cold War is won now. Probably be issuing by in traditional syrup. Finite. Full Spanx. There is one? No, but the value of effort. Thanks for years. Years zero finite one plus express to minus 40 before one case constant. No indication of one plus next system minus forties one excess to minus three on lane. See? No execute upper limit minus lower limit. So here are a problem. It ease in. Finance on Lord limit is sealed. And the quiet one now in final dress to Linus Lee thinks going zero one. Laws, steer olestra. Minus three people on this one. You okay? You ready? That's easy. Court one in case for three now, such case equals tree or giving function. There full. Thanks. So, uh, forthe next sees 21 blast next less to minus four. No, we need to find expected value. New. So new is Eun bi integration next to me. Clear for next T X. We'll put the radio people. Except for here For Texas to you one plus x stool minus four DX these constant. So take it outside of the integral. You're one plus access to minus force. We can write in Denominator. Now substitute tow you easy. 16 So actually, boggle your mind. This one now, technically, video system. So dairy radio off you steal. Very relatable. Funny, Siegel. And very radio taxis. DX from here. Do you report DX now it's up to you. All this time. He's into that. You're merely the quarter tree. Siegel Pulling finite. Here. Here we have X So accident Google, You minus one one. You just before do you Motsepe ready? In a minute. We have sealed room, mate. You divide it by us. Two for ease. Us two, minus three and one divided by U S to forties. You rest minus four. Do you dragging? Dig, Addition. So here you just remind store on my Mistral. My story is true minus three upon a tree. Now put back you is acquired one less x one plus extras to money's too divided by two plus one plus extra strong minus. Terry Dilated, they hear minus minus is going to be a plus. Spiegel in finite. No execute off limit liners. Lord limit three one fine. Understood. Minus two. Cycle laws one as you minus two. Divided by three minus lower limit is Siegel for one. Serious to minus two upon minus two. Loss one. Seal restroom minus three upon Lassen. Get by this. You have been fighting this room minus Dewey's seal. You're single again. Here in financial, still minus threes against evil. No one serious to minus two is one. So here's one like to one. Just remind us. Threes by three or C Euro Last three Almost one Michael plus one by three. So your music one might do so is some c Thank you.

We're told that in a week the number the random, variable X claims coming into insurance office has a poison distribution with mean when 100. We're also told that the probability that any particular claim relates to automobile insurance is 0.6 independent of any of the claim. We're giving her an invariable why representing the number of automobile claims. And we're told that wise then binomial X trials each with success Probability 0.6 in part a where has to determine the expected value of why give next equals X in the variance of why given X equals X So again we have that X is by no, really is a poison random variable the parameter of 100. And we have why given X Equals X is binomial Lee distributed right Parameters X and six in particular. This is going to be small x here. So the expected value of why given x equals X. Because this is the binomial variable. This is simply going to be a number of trials and times Probability P, which is 0.6 times x. Likewise, we have that the variance for a binomial distribution is N times P times one minus P which in this case is going to be X times 0.6 times 0.4. So we get points to four x in part B were asked to use part A to find the expected value of why we have the expected value of why given x from part A is going to be 0.6 x distribution and from the law of total expectation. It follows that expected value of why is the same as the expected value of the expected value of why given X. This is the same as the expected value of 0.6 x, which by linearity this is the same as six times the expected value of X and we're totally expected value of X could poison with parameter 100 is 100. So that's his point. Six times 100 or 60. Finally, in part c, we're asked to use part A to find the variance of why so again from part A. We had it the expected value of why given X was 0.6 x and the variance of why given X is 0.24 x and again because excess poison ran variable, we had the expected value of X is equal to the variance of X is equal to 100 the parameter now by the law of total variance, we have that the variance of why is equal to the expected value of the variance of why given X plus the variance of the expected value of why given X And again we have the expected variants that why given X, this is going to be point 24 x and expected value of why given x be determined was 0.6 X then using linearity for expectation, we get 2.24 expected value of X and using imaginative variants, we get 0.6 squared times the variance of X. As we said before both of these air 100. So we have 0.24 times 100 plus 0.6 squared times 100 which is 60

This question is about the poison telegraphic process, which is a process with a random variable and AT T and the way the process works is that at time zero and that he is assigned either plus one or minus one randomly, with probability, 50% each. And then events occur according to a post on process, and each time an event occurs, the parody of an itty switches. So, for example, if entity is plus one and an event occurs, then it switches to minus one and vice versa. We're also told that the probability of an even number of events occurring in the time interval from zero to t is 0.5. And so for part, A were asked to explain why for time greater than zero, given this statement here were asked to explain why the following is true. So if we look at the first of these two, it's explained why. The conditional probability that n A T equals plus one given in at zero is equal to plus one equals p. And remember, P is the probability of an even number of events in time zero to t. So if we think of our time interval. Let's say T is here. And as events occur according to a plus on process, that same event happens here. That's we'll call this capital T someone for the first event t's up to for the second and so on. So we could say that T I is the I've event in the interval from zero to t. So then if and at zero is plus one then we have and at the time of the first event will be minus one and at the time of the second event, plus one, which means that N a. T is positive Onley for even I so for an even number of events. So that and that t will be plus one on Lee if we haven't even number of events. So this means that the probability of enmity being plus one given that we started and at zero is plus one is simply this probability here the probability of the statement being true which we know to be p the probability of an even number of events in zero to t. And now for the second statement, we can use the same logic. So this time if and that zero is equal to minus one. That's the condition then and at the time of the first event is it would have plus one. And at the time of the second event is minus one and so on. And therefore you can see that n a. T is plus one Onley when the number of events on the interval is odd numbered. You can say that if we start with an at zero equal to minus one in a T is equal to plus one only if there is an odd number of events on the time interval from zero to t. So therefore, we could say that the probability that entity is equal to minus one alright rather plus one given that we started and at zero equal to minus one is the truth of this statement and the probability of an odd number of events is one minus p. Remember that P is the probability of an even number of events. So therefore we have we have explained why these statements in part a are true. Uh huh. So now, for part B were asked to use our results from part A and the law of total probability to show that the probability that entity is equal to plus one is equal to 0.5. We're all time greater than or equal to zero. Using the law of total probability probability that entity equals plus one is expressed as the probability that entity it was, plus one given at zero. It was one times the probability that and it's zero equals one. And so this is the law of total probability because we're finding the probability of n a t one plus one conditioned on an exhaustive partition of and at zero so and at zero can only take on two values, either minus one or plus one. And so, if we condition it on both of those values or all partitions of an ID zero in this manner and multiplied by the probability of in at zero equaling that value, we arrive at the total probability for entity equaling plus one. One more thing that we should note here is that probability of an it zero equaling plus one and the probability of an 80 equaling minus one are both have, and that's by the definition of the Poisson telegraphic process, and therefore that gives us the probability of entity being plus one equal to so this probability and this. So that's this probability and this probability are both half. And this probability and this probability are the results from part A of this question. So it's straightforward to find the answer. At this point, we have half times P plus half times one minus p, which comes out to have then for part C. We're asked to establish the property. That function mean of a telegraphic process is equal to zero, and standard deviation of the process is equal to one. So we have. This is equal to the expected value, the telegraphic process, random variable. And this is equal to entity times the probability of entity for all entity. And so there are only two entities. It's either plus one or minus one. So there's plus one and we've established that the probability of Equalling plus one his half and it can also equal minus one with probability one half it was zero. Now we can write the variance of energy, So an a t you could be minus one minus the main, which is zero times the probability that entity equals minus one, and it could be plus one. So one minus the mean squared times the probability that entity takes on plus one. And this comes out to you one times one half plus one times one half equals one. And therefore, we can say that the standard deviation of the process is also equal to one because it is the square root of one.


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