I.I.D. question: density function f(x) = 3x^2, 0< x < 1, 0 elsewhere.

Steven G

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Let X1, X2 and X3 be three independent, identically distributed random variables each density function f(x) = 3x^2, 0< x < 1, 0 elsewhere.

Let Y = Max {X1, X2, X3}. Find P(Y> 1/2).

I am at a loss here.

I will (hopefully) show that I know what I.I.D. means by this example.

Suppose you have an urn with two balls in it--1 red and 1 blue. You pick two balls, one at a time w/o replacement.

Let X = the color of the 1st ball and Y = the color of the 2nd ball.

The two outcomes are {red, blue} and {blue, red}.

P(X=red) = 1/2 and (coincidentally) P(X= blue) = 1/2

P(Y=red) = 1/2 and (coincidentally) P(Y= blue) = 1/2

So X and Y are identically distributed.

Now if we know the outcome of the 1st ball we then know the outcome of the 2nd ball. That is, for example, P(X=red|Y=B) = 1\(\displaystyle \neq\) P(X=red)=1/2. So X and Y are NOT independent and hence X and Y are not i.i.d.

Since X1, X2 and X3 are i.i.d. I would think there is no max for them--after all they are identical. Am I missing something here????

So I assume that Y= Max {X1, X2, X3} = X1

Then p(X1 > 1/2) 1- p(X1 < 1/2) = 1- (1/2)3 = 7/8. My solution manual disagrees with this result.

However, if I compute p( X11/3> 1/2) = p( X1 > 1/8) I get the solution manual's answer (511/512)
 
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Let X1, X2 and X3 be three independent, identically distributed random variables each density function f(x) = 3x^2, 0< x < 1, 0 elsewhere. Let Y = Max {X1, X2, X3}.
Find P(Y> 1/2). I am at a loss here.
I am also at somewhat at a loss here. But that said these are true.
If \(\displaystyle A,~ B,~\&~ C\) are independent events each with probability \(\displaystyle p\) then
\(\displaystyle \mathcal{P}(A\cup B\cup C)=3p-3p^2+p^3\)

\(\displaystyle \mathcal{P}\left( {{X_k} \geqslant .5} \right) = \displaystyle\int_{.5}^1 {3{x^2}dx} = 1 - {2^{ - 3}} = \frac{7}{8}\)

Again, I have no idea if that is what your text/lecturer means??
 
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