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(10) Suppose {X1-Xn}areiid with mean m and variance s2 Let mn be the average of the X;'s. Find the covariance of mn with (Xi-mn) for i=1,n...

Question

(10) Suppose {X1-Xn}areiid with mean m and variance s2 Let mn be the average of the X;'s. Find the covariance of mn with (Xi-mn) for i=1,n

(10) Suppose {X1-Xn}areiid with mean m and variance s2 Let mn be the average of the X;'s. Find the covariance of mn with (Xi-mn) for i=1,n



Answers

Find the mean and variance of the normal distribution of statistics using parts (a) and (b) with $m(t)=e^{\mu t+\sigma^{2} t^{2} / 2}.$ (a) Mean $=m^{\prime}(0)$ (b) Variance $=m^{\prime \prime}(0)-\left(m^{\prime}(0)\right)^{2}$

Part a tasks us with finding the sample variance of our data set using formula 2.5 in the textbook. So heart a the sample variance is going to equal. So using equation 2.5, the sum of squares over and minus one. So in this case, we're gonna subtract the mean of our values from each one of our date of each one of our values and squared. And then and then together, that's how we're gonna find R sum of squares and an equation two point. But now there are five step mortifying sample variance in this way. Step one. Let's add all the values together that would be seven plus six plus all the way plus 13 all the way down sweet at all our values. Together we get 70 to important In step two, we must find the means. So that's all the values that together divided by the number of values in our data set 10 that's going to equal 7.2 in step three. We're going to subtract the mean from each one of our values in the data set. So we're getting a seven minus 7.2 is negative point to six minus 7.2 is negative. 1.2 and so on. Five minus 7.2 Negative. 2.2 and 13. Minus 7.2 is five point. So you get all these values. Step four, we're going to get our summer squares are numerator. You're gonna take each value that we got in step three, square it and then add it to the next Value square and so on. So that would be negative. 0.2 squared plus negative 1.2 squared all the way to 5.8 squared, and then we square all of them and add them. Together. We get 70 three point six now, in step five, we're gonna put that all together here. I'll put step five over here. I'm gonna put that all together. So our sample standard a sample variance is going to equal 73.6 over the number of values in our data set subtracted by warned nine, which is approximately equal to eight. Coin 18 And this is our sample variance. Now for part B, we want to find sample variance using equation 2.9. So say the sample variance is equal to. So in this case, the equation will be the sum of each one squared minus the summation. Swear over and a your and minus flung. Now to solve it. In this way, there are four steps. Step one, just like before. We want to add all the values together. So we did that. That would be 72 in step two. We want to square. We want to some each square 77 squared plus six squared all the way to five squared plus 13 square square mall. Add them and you get 500 and 92. Step three. Yeah, we want to know. But we want to get the numerator for our variance. You're gonna get R sum of squares by. Look at our numerator. We want to replace the sum of X squared with 5 92 minus right place to some of X in our numerator with 72 square over 10 and this will get us 73 point sex. Our final step is to put this all together or sample of areas is going to equal r sum of squares which in this case is Step three which is 73.6 over n minus one, which is not which is going to be approximately e 0.18 now apart. See, we need to find standard deviation. Part C. What? Let's see. Let's racket this off here so we don't get confused. Part C. Standard deviation. So that's the sample. Saturday. Aviation is equal to the square root of ERM Areas in this case are variants is 8.18 So we take the square root of the 0.1 eat, and that is approximately 2.86 and that is our standard deviation.

Were asked to show two things in this question. So the first is if X is a random variable with mean mu were asked to show that the expectation of X squared is greater than or equal to muse squared. So we have X and a variable with mean mu. So this is one approach, so consider X minus mu squared. Since this is a square of value, it is always going to be non negative, so it has to be greater than or equal to zero. So whatever random variable X takes on this value will always be non negative, and therefore the expectation of an expression that is always non negative must also be non negative. So this expression inside the expectation brackets is taking on random values, but they're always non negative, and therefore the expectation of non negative values is not is not negative. This is equal to the variance of X, so therefore, the variance is not negative and using the variance shortcut formula, this could be rewritten as following and therefore we can say this, which is what we're trying to show. And now for the second part. So the second part is to show that the expectation of X squared is greater than mew squared unless X is a constant. So we have already shown the expectation of X squared is greater than or equal to mu squared. So now we should show that this expectation of X squared is equal to new squared if and only if she should say just on Lee. If yeah, X is constant. So if we consider than to be equal and we can rearrange as follows So this is equal to the variance. So it may be very intuitive to you that if the variance is zero, then the random variable X must be actually be a constant because nothing is varying. We can write this out a bit more formally, so these two parts are equivalent. That's by definition of expectation. So now both of these factors in the summation are non negative. So any probability by definition is non negative, and any value squared is non negative. So we have all terms in summation, must sum to zero, and therefore this factor must equal zero for all possible values of X. And this means that X must always equal mu and me was just the mean, which is a constant. So therefore, X is the constant mu. So therefore, we're showing that the expectation of X squared is greater than mew squared unless access constant.

In order to find the range of the data said, given we must subtract the lowest value from the highest value in our set. So a range is equal to or highest value nine minus our lowest value to which equals seven. So the range of our data said, given its seven in part B, we must use formula 2.5 in order to solve for sample variance. So that means our sample variance is equal to R sum of squares. In this case is going to be our the mean subtracted from each one of our data values in our set square A her and minus one. Now there are five steps in order to solve for sample variance using equation 2.5, Step one, some all the values together. So that's two plus four plus seven plus eat plus nine, which equals three Step two. We calculate the me of the data set, so that would be 30/5 30 divided by five, which is sex in step three. We want to subtract the mean from each one of our data values So that B two minus six, which is minus four four minus six which is minus two seven minus six, which is one eight minus six, which is two and nine minus six. Interest story. They remember all of these values, when added together should equal zero negative for minus two is minus 63 plus two plus 166 minus 60 So we're all good there. Known Step four. We wanna calculate R sum of square. So we want to square each one of the values we got in step three and then add them together. So that would be 16 plus four plus one plus four plus nine, which equals 34. Now, in step five, the final step. We put this all together to get our sample variance. So our sample variance is equal to their sum of squares, which we got in Step four. Just 34 over and minus one will satisfy the number values we have our data set minus one, which is four. My mother's 14 So 34 divided by four is equal to 8.5 and this is our sample varies now part C will put up here. We want to calculate for the standard deviation of standard deviation is equal to the square root of our sample variance, so that's equal to the square root of 8.5, which is approximately to 0.9 to.

In order to find the range of the given data set, we must subtract the lowest value from the highest. So here are range is going to equal E, which is our highest value minus three, which is our lowest value. So eight minus three equals five. This is gonna be our range for the data. Second, now Part B were asked to find sample of Arians using Equation 2.5 in her book. Now, Equation 2.5 states of the sample variance is equal to the sum of squares, which in this case, is going to be the sum of X minus X r squared. So we're gonna take the mean and subtracted from each one of our data values in our set square it and then added together, it'll ones here n minus one and they're five steps in order. Self sample variance is well, step one. We must find all we must some all the values in our data set together so that six plus eight plus seven plus five was three plus seven, which equals 36. Step two. We want to find the mean over a data set, so that would be our some, which is 36 divided by the number of values in our set, which is six. The people sex. So army is sex. Step three. We want to subtract. We want to subtract the mean from each one of our data values. So that would be six minus six, which is zero eight minus six, which is to seven minus six, which is 15 minus six, which is negative. One three minus six, which is negative. Three at seven minus six, which is more now instead, four we want to square and add the is we want to get r sum of squares so and stuff four We're gonna take each one of the values square it and added together so that zero squared 02 squared which is 41 squared which is one negative one squared which is one negative three squared which is nine and one squared which is one which is equal to 16 Now step five. We put this all together to solve for our sample variance or several areas s squared is equal to r sum of squares which we got in stuff for 16 over end minus one now and it's six. So this should be divided by by which is equal to three point to now apart. See, we want to find the sample standard deviation of sample. Standard deviation is equal to the square root of our variants of our sample variance. So that means our sample standard deviation is equal to the square root of 3.2, which is approximately one point 79 and that's our sample standard deviation.


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