So we have given that a probability. X measures in ppm. Okay. And ffx that this pdf is K X squared. 200. Minus X rays to it were excellent is from 0 to 200. We need to find Very okay for this. Pdf Toby. Well, second, we need to find if company does not accept impurity about 100 then Probability of not accepting that batch and see the expected value. The the Syria off we call it is capital f affects and e where? Why the nose Total impurity, then in percentage. Then what is Syria off that? Why? Okay, so let's start with one. We know. Ah, function. Toby pdf. Okay, so it must be between 0 to 1 its value and summation or since we're using CDF. So then Sorry. Pdf, which is density function. So integration, minus infinity to infinity off ffx must be equals toe one. Okay, Now, if you look into the first condition. Okay, so we have two conditions. So look at this function. This is even power, right? Two and eight. So these two values will be positive. And k, we'll assume it is positive. Okay, so therefore this function is or the first part is satisfied. So we need to check the second part of calcula value off. Okay, Okay. So let's start so minus infinity to infinity wherever it exists. I'm writing next step since this affects is not saying for all these limits. So it is said this effort is defined for 0 to 200. It means forest value. It is zero. Okay, so for minus infinity to zero, this is zero. Then we have from 0 to 200 our function and then we have 200 to infinity again. Zero Okay, so this part is gone. This part is gone. So we're left with this one. And here we can apply property Since we cannot factories this power eight so we can apply a property. So have you. 0 to 200. Okay, times 200 minus x squared. And this will be 200 minus off. 200 minus x it D x. Okay, So what? Property applied zero to a F F X DX is equals to zero to a fr a minus X dx. Okay, so this gives us 0 to 200. Okay. Now, if we apply formula, yes. So this gives us 200 square, which is four in 10 days to four, minus 200 to 400 x plus x square. And this gives us X rays. Tow it the X. So, in calculating this integration, we get 32 in two tenders to eight. They were by three. Okay. And this is equals to since with this limited one. So we get case 3/32 into 10 days to it, and that is a value offer. Okay. Okay, So you're done with the first part. So for be it said, if X is greater than 100 then company does not accept that patch, right? And we need to find probability off not accepting that batch. So we'll find our integration. So probability for X. Greater than 100 which is not accepting the batch is 100 to 200 integration off our function, which is 3/32 into 10 days to eight. And then here we have 200 minus x whole square times X rays to it, as we did in the previous step. Right. So on this, calculating this value from 100 to 200 off this again using the same integration formula we get answers zero point five. Okay, so there is a 50% chance that the company would not accept the batch if the ppm visible 100. Okay, so we're done with B now. Part C is five. Told to find expected value. So expected values given us integration minus infinity to infinity X times f or fix d X. And since we know this function is not defined from minus infinity to infinity using a previous reasoning so we can write this is 0 to 200 x Times K, which is 3/32 into 10 days to 8, 200 minus X squared times X rays to eight DS. And again, this is a simple integration. Will characterize this multiply X rays to it and x so on solving this integration, we get our answers. 100. So the expected value He's 100 people. Right? Okay, so we're done with part C. Now party is to find the Syria or community distributive function. So we know it is integration minus infinity to x offer period. And we know function is not defined for minus infinity. It is different from zero, so we'll take it zero to x fo fixes K which is three or that you do into 10 days to eight. Then here we have 200 minus x squared times X rays to eight D x. Okay, so on solving this. So we get. Okay, so So this gives us eight point fight two into 10 days to minus 12. X rays to 11. Minus one point hi into 10 days to minus eight. Then we have plus 1.16 into 10 days to minus six then and this is X rays to 10. This is Exodus to nine. Then the next term is five point 25 into 10 days to minus three into X rays to eight. Okay, so this time is in negative plus 1.7. So the 1.5 into X rays to seven and to 80 X rays to six plus 336 double zero X rays to five minus 24 000000 So that is five zeroes. Okay. Sorry. Exodus to food. And this is a negative. Plus, we have 18 to 10 days to seven x cube. So the scissor capital F off x. Okay, hope you don't mistake make mistaking this values because it's little long values. And most of them we can approximate 20 like Then there's two minus 12 and there's two minus eight, and there's two minus six, and this all can be added. So that will give us a four x also leave. Okay, so we're done with part D. So now to calculate part E. Okay, enough apart, e, we're told Capital. Why the notes total impure routine percentage. Okay, so we know our pdf is given to us. Yes, Kate. Times X square, 200 minus. Exhale Square for X between 0 to 200. And this represent ppm. Right? So why is representing in percentage? So we know for X equals to 200 I will be one. And for X equals 20 I must be zero right for 200. That is a maximum people that is allowed. And that must be close to one person, 100%. So that can be represented by value one. And this is zero. Okay, so that is nothing but dividing this entire time by 200. Right? So 200 x by 200. And here 200 way 200. Okay, so this value is nothing but ever. Why, right. Okay, so we know why is ranging from this to this value. So we know a relation is wise X 0 200 or excess, 200 times. Why? So now if you write that, Pdf So we have a pdf s. Okay. 200. Why? Squared right? 200 minus for this is it 200. 200 y there is to it. Okay, So this is a period function representing the probability density function with why, between zero and one, which is but our probability. Okay, so this is a period function. Now we can integrate this. So what will have you taking this? 200 days to to and from year 200 to 8. So that becomes 200 days to then Kate. Times y squared and one minus. Why? There's 28 So this is it. Pretty? If so, what will be the city of Syria? Philby. Integration of dysfunction from We were integrating from 0 to 200 since we had a limit. That but now our wise ranging from 0 to 1. So we integrate from zero to X right for CDF and our function is same as this. Okay, White Square one minus y whole square. D y. Okay, Now the problem is we need value of K. Soto. Find k again. This condition is pdf night submission. So let me right in. Function forms of function from minus infinity to infinity. DX is one. So here we have 0 to 1, 200 raise to 10 times K y square one minus y resto a d I should be cost to one. So in solving this, we get our chaos for 95 off. 200. Raise too, then. Okay, so that is OK. And this isn't parts per million value. Right? Okay, so we're in our parts per million concept. Okay, so now So this is why. Right? So this on integration with value Katie's, We get 4 95 over 200. There's 2 10, and here we have wire is too 11/11 minus two times. Where is to 10/10. Plus why? It is 2 9/9. And this is a city of for Why were we represent probability or percentage off ppm? Okay,