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The Auid flow two pipes combine to form the outfall discharge factory: The concentration of pollutants in each of the Auid filled pipes are random variables and Y a...

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The Auid flow two pipes combine to form the outfall discharge factory: The concentration of pollutants in each of the Auid filled pipes are random variables and Y and contribute to the concentration of pollutants in the outfall Z according to 2 = 0.3X + 0.7Y a) (5 pts) Find the expected value of the concentration of pollu- tants in the outfall, Z, given that the expected value of X is 10 puv (parts per unit volume) and the expected value of Y is 20 puv_ (10 pts) The two pipes are in close proxim

The Auid flow two pipes combine to form the outfall discharge factory: The concentration of pollutants in each of the Auid filled pipes are random variables and Y and contribute to the concentration of pollutants in the outfall Z according to 2 = 0.3X + 0.7Y a) (5 pts) Find the expected value of the concentration of pollu- tants in the outfall, Z, given that the expected value of X is 10 puv (parts per unit volume) and the expected value of Y is 20 puv_ (10 pts) The two pipes are in close proximity leading to the random concentrations X and being correlated with COr - relation coefficient pxY 0.5. Find the standard deviation of Z given that the standard deviations of X and are 3 and 5 respectively: (10 pts) The outfall discharge is considered contaminated if the pollutant concentration in the outfall exceeds 20 pUv. Assuming that the random variables X and Y are normally distributed, find the probability that the outfall is contaminated_



Answers

The amount of gasoline $X$ delivered by a metered pump when it registers 5 gallons is a normally distributed random variable. The standard deviation $\sigma$ of $X$ measures the precision of the pump; the smaller $\sigma$ is the smaller the variation from delivery to delivery. A typical standard for pumps is that when they show that 5 gallons of fuel has been delivered the actual amount must be between 4.97 and 5.03 gallons (which corresponds to being off by at most about half a cup). Supposing that the mean of $X$ is $5,$ find the largest that $\sigma$ can be so that $P(4.97<X<5.03)$ is 1.0000 to four decimal places when computed using Figure 12.2 "Cumulative Normal Probability", which means that the pump is sufficiently accurate. (Hint: The z-score of 5.03 will be the smallest value of $Z$ so that Figure 12.2 "Cumulative Normal Probability" gives $P(Z<z)=1.0000 .)$

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,

Were given a log normal distribution with mean 10,281 and coefficient of variation of 0.4. For part, they were asked to find the main value and standard deviation of the natural algorithm of X. So one thing that's important to note is when we have the mean with e subscript X are the same for the standard deviation. It means that these parameters pertain to the original distribution before it's in log space. That is, for a log normal distribution if we're given mu and sigma these air the mean and standard deviation once the distribution is converted to log space or once we have taken the log of the distribution. But when we have the subscript X, it is before we have taken the log of the distribution. Now we're given formulas in the textbook. Now, before we can make use of these formulas, we should first find the standard deviation of the distribution. I'm just noticing that I have these reversed here, and this comes out to 4100 and 12.4 now to solve for the log mean and the log variance, we can begin by finding the expectation squared. So this is equal to e to the tomb you plus Sigma squared and then noting that these factors are the same in these two equations. So we take the variance of X and divided by the expectation squared. We have e to the Sigma squared minus one, and this gives us a variance of 0.148 or a standard deviation of 0.385 And now to solve for the mean you can use this equation, we take the natural law algorithm of both sides, and this gives us a mean of 9.164 So now we have our two parameters that characterize our log normal distribution. So excess naag log normally distributed with mean equals 9.164 and sigma equals 0.385 And this allows us to answer the questions that remain So the probability that X is, at most 15,000 is asked for part B. So now we can standardize it, and this is equal to the accumulative probability of the standard normal curve at one point 174 and that comes out to 0.88 Now for Part C, we're asked for the probability that X exceed its mean value, so that's u sub X can be re expressed this way. That's one minus the probability that X is, at most 10,000, 281. And now standardizing this comes out to one minus the cumulative probability of the standard normal at 0.192 and this equals zero point 4 to 4. Now the question. We were also asked why this probability is not half. It might seem intuitive that the probability that a distribution exceeds its mean is half, but that only applies for symmetrical distributions. Remember, this is a log when a symmetrical when a distribution of symmetrical it's mean equals. It's medium, but a log normal distribution is right skewed. That's so if this is the peak, we might have the median somewhere around here. But the mean is somewhere to the right of the median. And so this area to the right of the mean in this case has come up to 0.4 to 4. Of course, the area to the right of the rate of the medium would be 0.5 and next for Part B were asked for were asked whether 17,000 is the 95th percentile. Another way to ask this question is is the probability that X is less than or equal to 17,000 equal to 0.95 and this comes out to 0.933 which is not 0.95 So therefore we can say that 17,000 is not the 95th percentile, although it is quite close to the 95th percentile.

We're told that a service station has both self service and full service islands. On each island, there is a single regular unleaded pump with two hoses. We're told that we should let X denote the number of hoses being used on these self service island a particular time. And let y denote the number of hoses on the full service island in use at that time were given the joint probability mass function of X and Y in a table in part A. Whereas to find the probability X equals one and why it was one to do this? All we have to do is look at our table. We see that the probability X equals one y equals when this is the same as P of 11 According to our table, is 0.2 up in part beef. We were asked to compute the probability that X is less than or equal to one. And why is less than a quarter one? Well, this is going to be the probability that X equals zero, and why equals zero plus probability that X equals zero, and why equals one, plus the probability that X equals one and Y equals zero, plus the probability that X equals one. And why was one? As I said before confined, he's been looking at our table. This is the same as P of 00 plus p of +01 plus p of 10 plus p of +11 p of 00 is 00.1 P F 01 is point of four. He of 10 is 0.8 and p of 11 His point to adding these Together we get 0.42 Next. In part, C has to give a word description of the event set X equals zero, and why not equal to zero and compute the probability of this event? So if X is not equal to zero, and why is 90 quarter zero? Well, this means that at least one hose is in use at both islands. Yeah, the probability that X is not equal to zero and why is not equal to zero? This is the same as P of 11 plus p of +12 plus p of 21 plus pf to to after plugging in values from the table and calculating, This is 0.7 in Part D whereas to compute the marginal partial mass function of X and of why so partial mass function of X. We confined by something real probabilities because this is going to be awfully some across the top row we get 0.16 This is when X is equal to zero, some across the second row. We get 0.34 when X is equal toe, one semi across the third row. Yet 0.5 when X is equal to two. Likewise, by selling the column probabilities, we confined partial matter function. Partial mass function for why we have that this is going to be some of the first column is 0.24 So this is when why is equal to zero. Some of the second column is 0.38 when why is equal to one into some of the third column 0.38 when why is equal to two were asked to use the are marginal partial mass function of X to find the probability that X is less than or equal one. So we have probability that X is less than a quarter one. This is going to be the partial mass function of X, evaluated at zero plus the partial mass function of X, evaluated at one and adding together from the above functions we get that this is 0.50 Finally, in part, E were asked if X and Y are independent of random variables. Yeah, well, I noticed that probability of 00 according to the table, is one, but we have that on the other hand, partial matter function of X at zero plus times the partial matter function of why then why equals zero? Well, this is going to be 0.16 times 0.24 just clearly not 0.1. It's actually 0.384 Therefore, it follows that variables X and Y are not independent. They are dependent.

Yeah that's probably been given the following didn't say function. We like to find a few different things from now. They would like to find the expected value of this density function And so they expect a value of why means we need to integrate across all of our Y values why times or density function and so taking this internal we want the integral of y times 10. There's one minus y to the ninth going from 0 to 1. Mhm. And if we just use a graphing utility in order to evaluate this, this gives us our average value is 1 11. Yeah. Now and be we'd like to find the expected value of one minus wife. Yeah. So the expected value of one minus Y is the integral from 0 to 1 of one minus Y. Times are density function mm Once again if we use a graphing utility in order to help us out here on the senator. Listen, your votes are average value here is 10 elevens. Which would make sense if they expected by Why is 1/11 and the expected by of one minus? Why would be the complement of that? Which is $10 on seaweed. Like the variance of the random variable one minus Y. Now, in order to find the variance, the first thing we need to find is the expected value of one minus Y squared. To be the integral from 0 to 1 of one minus Y squared times are density function. So now we evaluate this and this gives us 56 And so the expected value of one minus y squared is 56 mm. Well, we know our formula for the variance. Does the expected value of one minus Y squared minus the expected value of one minus y square. Mhm. Okay, this is 56 minus 21st square and 56 minus 10. 11 squared gives us five over 7 26 and so our finances 5/7 26.


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