5) Suppose C caalom Va /alle has CDE Functi 04 FC) = 38 fr OLXLz Find Hbe eX Pected value Of...

REVIEW Refer to the figure below. If $\cos D=0.8,$ what is length $\overline{D F}$ ?
$\mathbf{F}$
$\mathbf{G} 4$
$\mathbf{H} 3.2$
$\mathbf{J} \frac{4}{5}$

To show that the probability P is equal to one minus five. Eat in the minus two. We're going to start with the fact that the way function psy is equal to uh, E to the minus are over a divided by the square root of pi A to the third way is Boris Constant. So probability by definition, is equal to the integral over which we are the area over which we are integrating. Zero a of the way function squared times the four pi r squared because we're dealing spiritually here on the integration is over. De are so playing in our value for sigh and squaring it and pulling out the constants. Right? So four is just a content so we can pull it out of the integral. The A cubed is a square rooted, but he gets squared, the square root goes out, goes away slacking, get pulled out front, and the pie here gets canceled out with this pie here. Okay, so the integral then is zero a of r squared e to the minus. Two are over a So this is a pretty tricky integral. I went ahead and just looked it up in integration table. Eso doing that? We find that this is equal to I saw 4/8 to the third, um, r squared times a over to minus a squared are over too. No, minus eight of the third over four. Of course. Each one of these gets multiplied by e to the minus. Two are over a so we can go ahead and pull that out when we integrated from zero to a So we have to do within is clogging a for our and then subtract plugging in zero for our well when we plug in zero for our that's pretty easy, because the, um the, uh any value with or in it is going to go away. And remember that eat, the zero is equal to one. Okay, so plucking those in, we find that this is equal to we saw 4/8 of the third. You have eight of the third over to minus eight of the third over to minus stated third over four. Oh, all of that gets multiplied by each of the minus two. Oh, so that was from plugging in a for are now plugging in zero for our This is equal to plus eight of the third over for close off the bracket. Okay, so simplifying this, we see that this is equal to one minus five E to the minus two because the aid of the thirds canceling all the cases so that cancels the YMCA, multiply it through with all of these. That's what we were asked to show so we can box and then as their solution.

This question. We are given this circuit diagram before capacities and see. One is 10. My grandparents, C two, C three and C four, uh, have the same identical 20 micro fair capacitance. Um, we are given that to one. Is that the micro column? And I'm going to find the baby. Okay, so one thing we first need to do is to redraw, uh, we draw the second program. Okay, so, um, so this given circuit diagram is you have to Hello. Capacitors in series. I mean, two parallel, uh, network. Uh, yeah. Or 22 parallel capacitor network. And these two networks are in series. Okay. So we can actually redraw the and then one thing we need to know is that Okay, so for capacitors and, well, capacitors in parallel in the equivalent with capacitance is C one C two, just some there, Uh, capacitance is, and then in series, you have one over. We could learn capacitance if you go to one over C one plus one over C two. Okay, So the two parallel networks are the first, like one. We have, uh, the the micro Sarah key. And then the right one is 40 micro ferry. Okay. Okay. So, uh, we can you can have this background, and then we can start working on the rest. Okay. So you know that given Kyu Wan is that the micro column? So the one is, uh, q one over C one. We get three votes. Okay. Yeah, because everyone is 10 Micro fairy you want is to be micro column. So you put inside you get rewards. Okay, So you one and V two, uh, C one and C two are in parallel, so you want to be too? Are the same. Okay. We want girls to eat too close to three boats. Okay? So we can actually find Q two to be a C two e two. You get Andy, uh, micro fair at times, tree votes get 60 micro column. Okay, So he wanted you to the charge in Q one. Q two is the charge in this, uh, that the micro Sarah Ambassador. Okay, So you want to ask you to start inside here? Okay? The charge store. Mm. That the Microsoft right, Professor? Yeah, Secret, too. Uh, that, uh, 16 to 1980. My curriculum, uh, of church. Okay. And for capacitors, in series. Um, the charge are the same residents in series in the queue in 30 micro fairy capacitor is to go to Q 40 Micro ferret capacitor. So, uh, we at 14, my growth error capacitor B Q for the micro Farrah by See? Okay, So this is 90 Micro Colon. If I buy 40 Micro parent and you get, uh, 2.25 votes. Okay, So B A B is equal to will teach across the the micro Farah class, The teacher across for the microphone because they are in series. So we'll teach some some strategists. So you get three last 2.25 and you get 5.25 votes. Okay, Uh, in two sf, you get 5.3 boards. Okay, So the answer to SF is five point tributes, and that's all for this question.

Okay, so, uh, the problem We have the circuit. But you during next city he was in here is actually the extra one girl we have. No, but here 12 and different one. Giving you problems. Uh, my 4 30 uh, five micro Because we can see from the circuit that the wine is here. So suppose the combined capacitance off one to find and we would have radiation. He went prime equal to one over prime people to over one. Okay, so just plugging the radio one, too. So we can find out the one crime one over or over three. Piper Baird again. And, uh, you on prime? Exactly. Parallel three. This case to combine the capacity of the whole system people you want Crime Club three directly. Yeah, equal to three. 22 over three there. Yeah.

To find the value of sea, we will set the integral of X equal toe one, you know, evaluate on the boundaries from 0 to 4 of the function to C E to the negative c x dx. Evaluating this integral left with one equals two C over negative C E to the native c X evaluating at four and zero plugging in the foreign zero at X. We also see that this will cancel out in these seas. Here. Simplifying one equals negative to. Since those sees cancel out e to the negative four c but plugging in four and plugging in zero e to the zero power equals one Further simplifying this step have negative 1/2 plus one equals e to the negative four c And as we look to isolate, see to set that equal, we'll simplify 1/2 equals e to the negative four c And now to isolate that see, we can apply the Ellen function to both sides. So Ellen of 1/2 eagles Ellen of E to the negative four C, which is just negative foresee, and by dividing by negative four were left with the final answer of C equals negative. Ellen of 1/2 over four