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Use 360 amounts: Joore d] than Assume 1 the payout 306 sebiec tosses amounts 8 9 S300 receive a payout of S300 a S300 limit: follow solve - estimate - the 365 xsore...

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Use 360 amounts: Joore d] than Assume 1 the payout 306 sebiec tosses amounts 8 9 S300 receive a payout of S300 a S300 limit: follow solve - estimate - the 365 xsorestial final the parameter equation. and 340 the HL insurer observes who Suppose method of moments 300 the experience a following payout loss insurance company Use amount pays

Use 360 amounts: Joore d] than Assume 1 the payout 306 sebiec tosses amounts 8 9 S300 receive a payout of S300 a S300 limit: follow solve - estimate - the 365 xsorestial final the parameter equation. and 340 the HL insurer observes who Suppose method of moments 300 the experience a following payout loss insurance company Use amount pays



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Solve each problem. Maritime losses. The amount paid to an insured party by the American Insurance Company is computed by using the proportion $\frac{\text { value shipped }}{\text { amount of loss }}=\frac{\text { amount of declared premium }}{\text { amount insured party gets paid }}$ If the value shipped was $\$ 300,000,$ the amount of loss was $\$ 250,000,$ and the amount of declared premium was $\$ 200,000,$ then what amount is paid to the insured party?

Yeah. Okay, so we have this probability distribution function that gives thie benefits. Ah, received on a loss. So the distribution function is given by Teo over. Why Cute, then why? It's bigger than one and zero otherwise, and now we're told the benefit limit is ten. So R benefit. Why? What's just equal to what the benefit is equal to? Just the loss. Why, if why is less than ten? But then if I have lost more than ten, I'm only getting him back. So it's, uh, just ten. If I was bigger in time, and now we want to find he had expected about you. So that's just the inner girl. Okay, from well, here we're starting at once at one to infinity, But I need to break up when my variable is just why itself and when it is actually equal pretend so go from one to ten and then we have why times two over by cute And then when wise bigger than ten. This is actually just ten times to rely. Cute. Okay, so this is a factor to out one to ten of why two seconds and then I can factor a twenty out and I just have wider the minus turd. Okay? And when we get here, Well, this is negative, too. Why did the minus one evaluated from one to ten and then minus s o? It's twenty. And then I'm gonna end up dividing by negative two Sundays, ten wide on my second. Evaluated from ten to infinity. So this is negative. Two times one turf. Nice. One minus ten times. That's going to give me zero fine. Sworn over one hundred. So this it is negative. Nine tensing a set of five over nine. Plus, it's just going to be positive. One tenth, which is sees a B fifty plus nine. Fifty nine over ninety. That's the day every time. And this is no equal to about Oh, I think No, this is nine over five. Okay, so we just against this is nine over ten Negative. Nine hundred ten times negative, too. Yeah, that's positive. Nine over five. I just got my faction fixed up, so Yeah, I kind moved five. Okay. And this is going to be eighteen over tens and nineteen over ten. Never go. That looks like one of the injured choices that looks like cancer Choice City

Okay, So for this problem, we've got a density function for a loss. Why? Given as to over y to the power of three for all. Why greater than one arrest? What? The expected value? Uh, of pair of some policy is based on losses which are based on this density function. Ah, the policy has a basically a stop loss at 10 at a value of 10. Meaning that, um, for everything up to 10 the insurance policy will be paying out the amount. But once you reach the 10 3 insurance policy only pays out a value of 10. So it kind of works opposite of how a deductible works. So we have a couple of ranges of losses. Why that interest? Us? We have our losses from one up to 10 and we have our losses greater than 10 for everything from 1 to 10. The payout from the insurance company is why right? As long as our losses less than 10 the insurance company pays out all of it. Once we exceed 10 the insurance policy only pays out the maximum value of 10. So when we are trying to find the expected value, you know We're kind of summing up all of the ah payouts times the probability of those payouts. And since ah, payouts air continuous, right, We could take any value. Uh, this summation becomes an integral um, And since we have two ranges, we're gonna have to inch girls. So the 1st 1 looks like the girl from 1 to 10 and the amount that is paid out at this particular time is why and the probability of each individual. You know, if if we want up out of two and we want to know the probability of two, we plug it into this f of why. So if we want the probability of any of thes particular payouts happening, we multiply it by the, uh, density function. So here we have our payout in our probability of our power for this interval for the interval of greater than 10 we have a similar case, except that are payout now is always 10. So instead of a y here, we're gonna have a 10 times the same density function two over wide 1/3 cause even for why that are greater than time, this still tells us how likely they are to occur. So if we evaluate thes into girls, this 1st 1 thes wise will cancel out. We have too wide to the negative to power. And so that integrates out to negative two y to the negative one. And this is gonna be evaluated from 1 to 10 and we're gonna add over here. Similarly, we're gonna have a y to the negative to power, and we're gonna have a negative 10. That's that constant in front. And this is gonna be evaluated from 10 to infinity. Okay, So if we go ahead and plug in our values over here, playing in the upper bound of negative or of 10 we're gonna get negative too, over time, minus negative to up here. So this could be positive over one. That's this guy. And if we do the same over here when we plug infinity, we're gonna get infinity on the denominator. A constant on the numerator. That means we have zero. The whole term goes to zero, and then the second part, we're gonna have 10 over 10 square, which is 100. So all in all, we've got to minus two tents plus 1/10 so we should have tu minus attempt, which should be 19 10th or 1.9 as a final answer

Words hold on. A 55 year old man deposits $50,000 with to fund ingenuity. With an insurance company, the money is invested 8% per year, compounded semi annually. And then we're told that he is to draw some annual payments until he reaches. So until he reaches 65 years, what is the amount of each of the payments? Okay, so first we know that N is going to equal 10 years because the 65 miles 55 is 10 and we know that its semi annual. So our end is gonna be the 10 years times two payments he's making semi annual, which is gonna be 20. So now our end is actually gonna be 20. Okay, then we know that our annuity is going to be $50,000 and we're also look at our interest, which is that 8% yearly. So what we're gonna do is take the 0.8, which is what it would be as a decimal divided by two. His semi annual point. Oh, for so our new I this point before So now that we have that, we're gonna look at our equation to solve which we know is our annuity is what we're investing's or payments one plus our interest to it and power know ahead of myself, There to the end, minus one. And that is gonna be over. I So we do this. It would be 50,000 equals R which we're still trying to find one plus the 10.4 to the 20th minus one over point. Oh, for so now that we have this, what we can dio is rearranged. We get our equals 50,000 times 0.0 for over our 1.4 to the 20th minus one. So then are is going to equal 2000 over 1.19 When we do the math, then we know after we do this are are is going to be 1680.67 So then we know that the amount of each payment to each payment is going to be $1680.67. And that's our answer

Number 35 took me a very long time to complete, but I think I finally got it. So gotta be all feminine with entrance policies. Um, first thing we know is that there is a loss which is uniformly distributed between want between zero and 1000. So what is, um, the payment that we can expect from this loss? Well, this problem is heart digressed. Um, even when I re read my notes, but we'll get through it, I promise the expectant payment in terms off the loss. Um, well, if the loss, if if there is a deductible here, that is between zero 1000. If the loss is higher in the deductible, then the insurance company must pay. But if the loss is lower than the deductible, then the insurance company doesn't take anything because the client has to pay the deductible. Right. So our payment, the payment from the insurance company, um, I will read it as a function of the loss equals zero. If, um, if the losses between zero and D and it is equal to x minus T, if the loss is between the and 1000 why X minus D? Because it will be okay. Um my ex. I mean the loss. So if the if the loss is higher the deductible, then the company, then the client. Okay, let's start again. If the loss is hiring deductible, then the client pays a deductible and the entrance company has to pay the remaining, which would be the lost. My necessity. Now, life on helpful for this is to try to find a MME Probability density function for the payment. What probability that I that the insurance company is gonna have to pay something, and then once we have this problem, the function we conducted the expected value for 30 average. So this is the probability, and this is the payment. Um, with the maximum payment they can make is 1000 minus injectable. They're never gonna pay more than 1000 minus in vegetable because the loss cannot be more than 1000 and the minimum that could be a zero. But if the loss is lower than the delectable V P zero, so most in most cases they're gonna pay zero. So here at zero, we're gonna have a very high point. And then, um, after that, there is a chance that they pay here. There's a chance here to the chance to be here. There's a uniform probability that they paid between D and 1000. So then they would be uniformed probability here. I'm gonna draw an empty circle here to say that at zero, we have to look this point here. Okay? Where to place those now? Um, well, the probability since since the last is uniformly distributed, there will be probability of D over 1000 that the payment zero right, that if that makes sense because the loss could be anywhere between zero and 1000 with equal chances. So the probability that it's slower than D is the over 1000. No, How to know? What's the value of this probability? Well, one thing we have to one thing that we know is always true is that total probability is always one. So the problem of you that remains here, which is so this was probably was the over 1000 it probably it. It's not the over 1000 with you. Well, we could calculators one minus you over 1000 because the sun has to be one. So here we have one minus the over 1000 which we have to divide, man. Yeah, I'm trying to find easier way to explain this, but I can't, um we have to divide by 1000 minus T. Why am I defending 5009 D you may ask. Well, okay, let's see. I want this area of this. I want the told probability off. Any payment to be one, right? That's always true. There must be a probably of 100% that any payment is made from 0 to 1000 dynasty. Now, if I think the area of this plus this value here, it's got equal one one equals. Do you ever 1000? That's this one here. Plus, um, the area of this that's a rectangle. So the area of this is 1009 sti here, times one minus the over 1000 divided by 1000 minus D. And then that's when those things came out. And then, you see, the equation is true only because we use that probability. Okay. I realized that it was a big bit backwards is an explanation. So I would trade again. We thought restarting the whole video. I'm gonna try. You're starting again. Why? We chose this probability here let's say we don't know what that is because we don't feel, you know first, um, and let's call it I don't know. Oh, my God. What did it? Let's call it each. Now, that's serious. All that stuff. So which it was a big Guerry, Reaser. I knew I raised this stuff. I'm going to start this explanation again. Um, okay, that's fair. So we're gonna start from the idea that this point here is that the over it does. And I think that was pretty clear. Um, because there's d over 1000 from abilities that the payment is zero. So here payment zero probability is delivered. 1000 now, for the rest, we've, uh We've made clear that the probability should be uniform. Um, but it's gonna be lower than the payment zero, because again, looking at this graph way, see that there's a lot more cases where the payment zero. Then when the payment is Oh, no. 10 are 20. You know, a few men of 10 and he has 1.11 less where it can happen and the payment of zero. There's all those possibilities now. We knew the tour probably has to be one. So one equals the over 8000. That's this point. Plus this red area, just a JJ times 1000 minus t. And that's how we find age we can solve for H here, huh? So you can do your job right here and realize that which must be one minus the over 8000 divided by 1000. My honesty. Okay, so now we have this probability density function for the payment was the expected value. The expected value is the average. Yeah, and so we're gonna integrate over all possible that I use from zero 2000 minus D. And the equation for the average is x times f of X and f of X. Is this thing here, right? The pro very density function. Since there's rupture here, we have to do two different cases. But the good thing about it is that at this point here, which is highly likely x zero. So you would have zero times the over 1000? Doesn't matter it zero. So this green point here, we don't even care about it. In the calculation of this expected value. Now we're gonna integrate over here, and what we get is that, um f of X for this area is equal to, um beach. So one minus the or 1000 divided by 1000 dynasty and then solvents and drool convict the constant out. It's a very long problem there. There must be a simpler way to to do it. But I'm sorry I confined it. This this is one way to do it. That that would be well for sure. Uh, there must be an easier way now. I'm integrating x from 0 2000 minus Dean. Right. Rewriting the constant on DDE extend you of excess. X squared over two invalid from 0 2000 minus d and what you get. It's same old constant here on DDE 1000 minus D squared over two minus zero. Yeah, right. Let's of things you can simplify here. We can remove this. We can remove this, um, this zero matter. So let's read that on new She new equals one minus the over 1000 times. Um, 1000 dynasty. So? So that's actually, um 1000 times one minus the over 1000 square. Because I took it 1000 out of this and parent business. And then I had the same thing is here, so I could square it. Okay, here's Mu nu is expected value for our payment. So on average, our payment is gonna be this now, in the problem, they want to set the deductible so that the expected payment is 25% of what it would be with no deductible. Okay, if the equal zero than the expected payment he calls. Well, 1000. So what we want is to set D such that expected payment is 25% of 1000. So $250. Right city. So 250 equals 1000. Andi going to use this equation here, which is the equation for the expected Raymond one minus the over 1000 squared. Um, and when you solve that for D, you will find that Deep Peoples 500 500 dollars. Yeah, So that was Ah, really hard problem. I don't really see how it fits with the other problems which I found what easier in this one? Um, it's I think there must be an easier way to do it. But I couldn't fire that I spent like a good hour in this problem. So the first time, as much time with them read to get on for this problem, sadly, but at least you have one way to do it. I think the key seem to understand this problem is to find a probably density function for the payment, starting from the probably density function for the loss. And then from there, once you have that, um, it's a lot of math, but all this matter was very reasonable for you guys. I'm sure you can do all this math. The hard part was really How can we find probability, Density function, foreign payment? Um, yeah. Hope for you could fall true, but


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