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Let $X$ be a nonnegative continuous random variable with cdf $F(x)$ and mean $E(X)$ .(a) The definition of expected value is $E(X)=\int_{0}^{\infty} x f(x) d x .$ Replace the first $x$ inside the integral with $\int_{0}^{x} 1 d y$ to create a double integral expression for $E(X) .$ [The "order of integration" should be $d y d x . ]$(b) Rearrange the order of integration, keeping track of the revised limits of integration, to show that$$E(X)=\int_{0}^{\infty} \int_{y}^{\infty} f(x) d x d y$$(c) Evaluate the $d x$ integral in (b) to show that $E(X)=\int_{0}^{\infty}[1-F(y)] d y .$ (This provides an alternate derivation of the formula established in Exercise $38 . )$(d) Use the result of (c) to verify that the expected value of an exponentially distributed rv with parameter $\lambda$ is 1$/ \lambda$

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

University of North Carolina at Chapel Hill

Piedmont College

Idaho State University

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all right. Now they have given it an expression for expectation effects. We're starting with the eye, but right now, so expectation off X in this particular problem is given as indicative. Go to infinity. This is X into F off X B X. So in the first month, their goal is to replace this ex tone with an expression. So I want to realize that part here. Someone to replace expert and giggle zero through X. This is one in tow, Levi. And if affects the extra meaning as it is. So if I heard this expression with proper arrangement of terms, the final expectation expression that I'm getting now is one into FX will give me FX. And now the order of indignation is due by the excellence is with the first part. It's not moving on to the second bark. They're told that we actually have to Now, in the order of this integration from, say, the vile e x to d x d vile. So I'm gonna force consider that given limits that we have here, you can see that the elements are made up functions off eggs, so including them. Equally wise, we have eyes from zero. Why is equal to X and outer limits? The homeowners X zero x equal Do infinity by turning point us on the next y axis. So I'm thinking this day that I'm here. It's an exact science. This is by access the sir prisons they questioned by Kordell zero two separate since the question X equal to zero. Okay. And we are viable to accept his lying passing Trojan. Okay, this is the origin. So no, we're gonna take a horse, understand? Because we're changing order to dig deeper if I take a strip like this. And if I shared that strip so we can see that from two starting from the line. Why quarterbacks? And here it is leading to infinity. And there's some lovely Why is going to be start from exactly Which is why I called you zero and this very cold. I hope you're infinity. So over here we can see that X is equal. Do. But on this side you can say that excess standing infinity. So the new limits that I will be getting now Okay, that will be first with I don't elements off X so excess from by tow X is equal to infinity on we have eyes going from zero. Why is equal to infinity? So if I needed with the new limits, the expectation of the expectation affects now can be two dinners so that limits of I will be writing art because their constant zero toe infinity and then I have integration excess from buy to infinity. The function remains as it is and we obtained order of indications of the X and B. Bye. So this is what Bill and Linda be part now moving on to the bar, See for the party. They have said that we're supposed to evaluate this inner integration which is actually the first or indignation with X. So if I even aware that I want to get a new expression for your fix so the outer limit remains as it is Now we know that small FX is the pdf and if I integrate this, we will get the CDF over so which is represented with a capital f affects. And so the indignation is done so I can put the limits from wired to infinity and do you so this part means, as it is now upon the limits I have. This is F off infinity minus f off riot. Anyway, Now we know that CD of the highest value possible is actually one because it ranges from 0 to 1. So we can therefore a place if often for knitting with one. So the final expression that I get as indictable zero to infinity in the bracket we have one minus f off. Why do you buy? So this is what is the expression for them Parsi now moving on to the next part for the Barney. There have been supposed to take the expression that we have derived Enciso, which is I just realized that you affects is integral zero to infinity black we have on minus the CDF off by and Dubai. Okay. Now they're told to consider the expansion the solution with parameter lambda and go take the CDF offered. So we can therefore write f off by the CD If the parameter lambda definition is one minus used to minus lambda Why? This is an advise ready than or equal to zero. I had zero in any other case since our indignation is from zero to infinity will be considering the first value so successful that in the expression off your eggs. Okay, so this is equal to individuality or to infinity. We have one minds and first apps. You'll devalue off for five. This is one minus is to minus lambda. Why? With diva? So we can cancel one here and minus and minus will become blessed. So I'm going to actually indignities to mine Islam there. Right. So you just minus lambda wise indeed. Nation upon minus lambda. And the limited strums you do infinitely. Not the nominee is a constant. I'm taking it outside in the upper limit for infinity. Here it is, two minus Infinity is zero minus E gs to zero gives me one. So therefore the final expression for expectation off X. When the parameters lambda it is given by one bite down. So we're done window. The part also

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