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Let $I_{i}$ be the input current to a transistor and $I_{0}$ be the output current. Then the current gain isproportional to $\ln \left(I_{\mathrm{o}} / I_{i}\right) .$ Suppose the constant of proportionality is (which amounts to choosing a particular unit of measurement), so that current gain $=X=\ln \left(I_{\mathrm{o}} / I_{i}\right) .$ Assume $X$ is normally distributed with $\mu=1$ and $\sigma=.05$(a) What type of distribution does the ratio $I_{0} / I_{i}$ have?(b) What is the probability that the output current is more than twice the input current?(c) What are the expected value and variance of the ratio of output to input current?

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

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All right. So this question say is that there's an input covered in our poor Clarence. I'm gonna write that park. You're so I still was denoting the output current, and similarly I want is denoting the input current. Okay, Now they have said that Ellen off I zero by I want is ah, normally, yeah, that's ah, normal distribution. So in this case, we can therefore say, since there is Eleanor here, So we can therefore say the description off the long armor functions sees that Oh, I was evil by Ireland and therefore bedridden as, uh, you can see it's a little normal distribution. Yeah. So since Ellen is there, we're here, and it's a normal distribution. So that is why we can say it's actually up. I hope I haven't is a long normal distribution. So this is the party that you've just proved now moving on to the part B. They have pulled us. Ah, refuse the formal off CDF for long normal function and find the probability there I zero by Ivan exceeds two. So for the B park, I want to say, uh, using the formula off CDF. Okay. And this is for Otto no normal distribution. Okay, so then we can apply the probably logic. You're that probability off I zero by I want. If it is greater than two then I can write this as one minus probability off I Is it a lie I want is less than equal Toto, this particle apart. I can replace this. Buy the CD a function here. So then this switching to one minus. I can write this as f off the black A tub. We've been given the valuable here. We're finding it for Britain And so she just will become the X Men Do that have been given here or the eyes you know, by Ivan Well, you can see And, uh, this is going to be mule as a force parameter And the next parameter is Sigma. So if I substitute devalues overheard which is given to us in the question this will be one minus f off the value for X that we're finding is too. And me was given as one and take what is given as 0.0 fight. So this is given by one. Mine is now the CD African. Did I hear you place? This meant respected that area. Logic as this is Ellen off to minus one. Born a 0.0 fight. So this value that I've got is based on the logic of this is X minus mu upon Sigma, just there for the standard normal. Brilliant. Which is they? So I can say the reason applying this year is that Zed is given by X minus mu upon Sigma X Over here is Ellen off two over here. Okay. Folder if I simplify, this party was in the last letter. Begin this as one minus five off minus 6.13 Not from the table. We're supposed to look this value, but if you see this value from the table, it's a very small value on DA, which is actually Ah, if you use an approximate value, we conceive is approximately going to be equal to zero. So I will therefore say that you know, fi off minus 6.13 is approximately going to people do zero. So I can therefore say that probability off eyes eoe buy I want which exceeds pool it actually hold one minus zero, which is Wow. So this is the second part that they have asked us not moving on with our part, they will find the expectation for a zero by one and also the variance using the low normal distribution formula. So going further to find out the expectation the part see, yeah. So the expectation off X for a normal distribution. This is given by Ely's to mu plus Sigma Square by two. And, uh, the formula for villians off X is given by e days too. This is stool New Bless Sigma Square. On the bracket we have heat is to Sigma Square minus one. So if I used the values that we've been given Mu and Sigma so fast after that over here a small change to eat is to one. Plus, this is 0.5 the square divided by two. So if you put this expression, the calc down so that you would get is 2.7. This is the value off your effects. But actually excellent. Sure for this particular summer's eyes you buy, I want. So this is the answer and we'll hear your finding variance for eyes it'll buy. I want. So substituting the values I will have eat is toe to toe one will be two. Plus, this is 0.5 square. I'm in the bracket. I have CDs to again 0.5 square minus one. So again, if you might apply and simplify this using the calculator, the rather than we will be heading is 0.185 So, uh, these are the expressions that they had asked us to find.

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