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Question 5. Consider the Kolmogorov-Smirnov test statisticDerive the cumulative distribution function (cdf) of the uniform distribution in [a, 6]...

Question

Question 5. Consider the Kolmogorov-Smirnov test statisticDerive the cumulative distribution function (cdf) of the uniform distribution in [a, 6]

Question 5. Consider the Kolmogorov-Smirnov test statistic Derive the cumulative distribution function (cdf) of the uniform distribution in [a, 6]



Answers

Use a symbolic integration utility to find the mean, standard deviation, and given probability. $$ \text { Function } \quad \text { Probability } $$ $$ f(x)=\frac{1}{6} e^{-x / 6} \quad P(x \geq 2.23) $$

So we found out from one of the last problems that the F of zero is one third And that's also equal to the f. Of one, And then the f of 1.5 is equal to the F of two, which is 16. Now, when we want to find the cumulative distributions, so the ffx at zero is going to be 1/3, and then the ffx at one is going to be one third plus one third to get us two thirds, and then the ffx at 1.5 is going to be those two thirds up there, +16 to get us +56, And then the FX at two is going to equal one.

57. They want us to find the 15 percentile for the sums. So use inverse Norm White 15 for my area 400 for my meeting and 8.16 fine, my standard deviation. And when I do that, I get really 91 play by 38

For this problem, you're going to use the applet for chi square probabilities and quantum files to find the probability that the sum of Z sub I squared Is less than or equal to six. And we're going to let I go from the values of one 26 and you need to recall or hint that the sum of Z squared sub i As I goes from 1 to 6 has a chi square distribution with degrees of freedom equal to six. So when you open up the applet you're going to see a chi square distribution and your chi square distribution is a curve that is skewed to the right. And at the top of that you're going to see a location where you can put in the degrees of freedom. And at the bottom you're going to see two locations, one where you can put in an X. Value and one where you can type in a probability. So since we are trying to determine a probability, we're going to leave this one blank and let the calculator or the applet generated. So we know that our degrees of freedom are six. So we're gonna put a six in here and we are trying to determine the fact that Z squared is less than or equal to six. So six is going to be right here on the curve. So we're going to put a six in here and remember what we're looking for is we are concerned with being less than six, so we are concerned with this area right here. Unfortunately, though the applet doesn't go and give you the area to the left instead, it gives you the area to the right. So when you're looking at your applet, you're going to see the area over here shaded in And it's going to generate a value and the value it's going to generate is going to be a .4-3. So that is telling you that the area right here is .4-3 and we know that the Curve, the entire curve, the area under the entire curve is equal to one. So we want this side, so we know that the side were interested in plus what the applet just gave us has to equal one. So therefore what we are interested in will be one minus 10.4 to 3. So we could say that the probability of the some of Z sub I squared as I is going from 1 to 6 being less than or equal to six will be a probability equivalent to approximately 60.577 Which is what we get when we subtract 0.4 to three from one.

It is a problem. FX is X over eight and three is more than X more than five. So this is the left side. Then we can calculator. The integration. So it's from 3, 2 x. And this is X over eight. The X. So first we calculated integration of X over eight which is four over X square. Oh no, this is 16 over square. Yeah. And it's going from X to three and the answer would be x square over 16 -9 over 16. Yeah. This is important because with this coefficient, this constant, we can make sure that the CDF is uh it's a probability distribution function. So it's zero. When X. Maury could ansari it's X squared over 16 minus 9/16 When three is more than x rays is more than five And this one when x Ecuador record and five. So that would be the answer.


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