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Let $ f(x) = 30x^2 (1 - x)^2 $ for $ 0 \le x \le 1 $ and $ f(x) = 0 $ for all other values of $ x $.(a) Verify that $ f $ is a probability density function.(b) Find $ P(X \le \frac{1}{3}) $.
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Calculus 2 / BC
Chapter 8
Further Applications of Integration
Section 5
Probability
Applications of Integration
Harvey Mudd College
University of Michigan - Ann Arbor
Idaho State University
Lectures
03:12
Let $f(x)=30 x^{2}(1-x)^{2…
05:45
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Let $f(x)=x e^{-x}$ if $x …
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Verify that each of the fo…
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Consider the following exp…
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01:42
All right, so we are given this function f of x, and the first thing we like to show that f of x is appropriated to the function, which means we need to show 2 things. The first 1 is to show that f of x is not actual function, which makes f of x, is 4 equal to 0 for o x. Secondly, we have to show that the integration of f of x, dx from minus infinity to infinity is well. The first 1 is there for because the only thing we have to worry about is this x forward 2 and a minus x squared, but there are square functions, so both of them are great or equal to 0, which means 30 times x. Over 2 times minus x to to quarrel to 04 x- and this simply implies that our f of x is a non negative function, so f of x is or equal to 04 x and secondly, we have to show that this integration is 1. But if we look at this function again, the only known 0 peace of this function is 30 times 2 times 1 minus x to 2, and this is what x is between 0 and 1 point. So this integration is actually so. This integration for minus infinity to infinity of the x in our ward case, it's just the integration from 0 to 130 x through 2 times, t minus 42 x and they see the energy repeats, and this upper and lower limit totally depends on the interval of this X, all right, so let's evaluate this integration so because 30 is a scaler we can take 30 outside and what i'm going to do now is to expand the bracket 30 times. Integration from 0 to 1 x equals 2 times 1 minus 2 x, plus x, 2 x, and then this is x for over 2 minus 2 x, cubed plus x to the power 4 and then by the general rule of integration. This is 30 times a third of x, cubed minus 2 over 4 x to the 4 plus 1 over 5 x to the fifth and we're going to evaluate this from 0 to 1 point so plug in the upper limit. We have a third times 11 over 2 thes, 1 old tree, just this 2 or 4 times 1 plus 1 over 5 times 1 and minus 0, because we'll plan the lower limit 0, the formula we've got 0 and you say that they say it is 30 Times 1: over 13 point, so this is 1 right, so for part b we would have to find what's the probability of x, smaller or equal to a third, so again by declaration. This is just the integration from 0 to a third f of x, dx and again, the only numero piece of f of x is when axis between 0 and 1 point. So this is simply the integration from 0 to a third 321 minus x to the power 2 p x and from part 1. We just found out what the integration is. This is 30 times the third x cubed minus 1 over 2 x to the fourth plus 1 over 5 to 5. But in this time we are going to invitin this function from 0 to a third so again plug in the upper limit. We have a third time, a third cube minus a half times a third to our 4 plus a fifth times a third to poor, 5 and minus a big 0 here. Because, while we pluck in the lower limits during the formula again 1. And if you use a calculator or some computer softer or say that this is a sorry 17 over 81 point,
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