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Let F be a random variable following a f distribution with 7 &11 d.f,find Pr (f1,41 <0.73 A process IS producing parts where the thickness of the part is fol...

Question

Let F be a random variable following a f distribution with 7 &11 d.f,find Pr (f1,41 <0.73 A process IS producing parts where the thickness of the part is following & normal distributi

Let F be a random variable following a f distribution with 7 &11 d.f,find Pr (f1,41 <0.73 A process IS producing parts where the thickness of the part is following & normal distributi



Answers

Let $x$ be a continuous random variable over $[a, b]$ with probability density function $f$. Then the median of the $x$ -values is that number $m$ for which $$\int_{a}^{m} f(x) d x=\frac{1}{2}$$ Find the median. $$f(x)=\frac{3}{2} x^{2},[-1,1]$$

It's clear is that when you read here, So we're gonna use the means definition and we got from 0 to 4. One is X square DX, which is equal to 1/8 1/3 execute from 0 to 4, which is equal to 8/3 for the median we go from, um for f of x d. X is equal to one have. So we're going to calculate the first integral, which is from zero two em 1/8 x t x, which is equal to one over 16 m square, and then we plug this in to 1/2 and yet I am to be equal to two square root of two, and that's Armenian.

The expectation relax equals the integral of X time. It's probability density function. So this is an integral from one to aid X times 8/7 X squared. This equals the into from 1- eight. Okay. to over seven while we were at stairs and it's anti juridical functioning as 8/7 Law Gaps. We're Chico's 8/7 lager H, Which is approximately 2.377. The expectation of X squared equals the integral of X squared times It's probability density function. So this is from 1 to 8. 8/7 dx retrievals eight. So the various echoes eight times 8/7 log eight square, which is approximately 2.352. And a standard division is a approach approximately 1.25 34

Okay, so here I'm given my mean value, which is 120. And I'm giving my standard deviation, which is 30. I'm also given my endpoints, which means that I need to find the probability that X will be between. Let's see here. I believe it was 55 100. That's exactly right. 55 and 100. Okay, so I need to remember a couple formulas, namely, that p of I should say, the probability that X will be between A and B is F B minus. You over. Roque minus f of a minus mu Asia Row and F is denote ID by one over the square root two pi the integral from negative infinity to Z of Let me double check here E to the negative X squared over two. I'm sorry. Over to D X now. This is not uninterested that we're going to be evaluating. Okay, um, all we're trying to do is really solve this part, and then we're going to use a computer to integrate this. So let's see here. So first I need to find what my inside values are for F. Okay? And that will represent Z which is what I will plug in here. So first, my first value of F B minus mu over row is going to be yes. Of be 100 minus 120 over 30 which equals f of negative two thirds. Okay, so first I'm gonna use a calculator or computer algebra system or whatever you want to use. I generally for either a calculator or symbol lab from negative two thirds to negative infinity. This'll entire function right here. You need to the negative X squared over two DX. Okay, okay. And so that one will give you. Let's see that will give you about I should write this, actually, about 0.2525 Okay, so now I'm going to evaluate or find rather the next Z, which is a minus mu over row, which will be f of is 55. Oh, sorry. 55. Okay. Minus mu, which is 120 over 30. Which will be f of negative 13 6. So when we evaluate that again using a computer, you do not need to do this integral e to the negative. X squared over two t x. We're going to get approximately 0.151 So now I just subtract this. These two values I'll have 0.2525 minus 0.151 And when I do that, I will get Let's see here I will get approximately 0.237 four at


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