Author Topic: DCC'S SUPER AWESOME FUN HAPPY TIME SPEC QUESTIONS EDITION 1 (Read 1262 times) Tweet Share

dcc · « **on:** June 14, 2009, 11:50:37 pm »

subject says it all.

answers will be done if the community REQUESTS IT

THANKS, DCC.

enwiabe · « **Reply #1 on:** June 15, 2009, 12:00:47 am »

I have the answer to Part F: Prove that dcc is no life.

The proof is the attachment in the original post.

jk nice work dcc.

Cthulhu · « **Reply #2 on:** June 15, 2009, 12:01:55 am »

I SUSPECT EVERYONE WILL ENJOY DOING THESE.

dcc · « **Reply #3 on:** June 15, 2009, 12:02:35 am »

Any VCE student who can do question E is awesome. Officially.

Addendum: This involves both the death of enwiabe & the solution to the stated problem.

zzdfa · « **Reply #4 on:** July 02, 2009, 12:36:48 am »

E. was pretty fun,

(ln(64)+5)/36 ?

where'd you get the problem from?

edit: checked my answer on a computer, here's my working

dcc · « **Reply #5 on:** July 03, 2009, 05:43:38 pm »

I think Ahmad said it came from a probability textbook, but I'm not certain. He may have made it up himself

I took essentially the same path, but more stumbling around in the dark. For $a, b, c \in \left(0, 1\right)$ :

First off, owing to the fact that $b$ is uniform, we know that $\Pr(b < x) = x \implies \Pr(b^2 < x) = \Pr(b < \sqrt{x}) = \sqrt{x} = \int_0^x \frac{dv}{2\sqrt{v}}$ . From this, $\Pr\left(\frac{b^2}{4} < x\right) = \Pr( b^2 < 4x) = 2\sqrt{x} = \int_0^x \frac{dv}{\sqrt{v}}$ for $x \in \left(0, \frac{1}{4}\right)$ .

Note that $\Pr(ac < x) = \int_0^{x}\left( \int_0^1 dc\right) da + \int_{x}^1 \left( \int_0^{\frac{x}{a}} dc \right) da= x + x \left[ \log( a ) \right]_x^1 = x - x\ln x = \int_x^0 \ln(t)\; dt$ . This form arises from considering a few 'cases'. If we have that $a \in \left(0, x\right)$ , then we know that any choice of $c$ fulfils our requirements. Similarly, when $a \in \left(x, 1\right)$ , then we require that $c \in \left(0, \frac{x}{a}\right)$ .

Combining these we attain the following result:

$\boxed{ \Pr\left(ac < \frac{b^2}{4}\right) = \int_0^{\frac{1}{4}} \frac{dv}{\sqrt{v}} \left( \int_v^0 \ln(t)\: dt\right)\: dv = \frac{5 + \ln(64)}{36} }$

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Author Topic: DCC'S SUPER AWESOME FUN HAPPY TIME SPEC QUESTIONS EDITION 1 (Read 1262 times) Tweet Share

dcc

DCC'S SUPER AWESOME FUN HAPPY TIME SPEC QUESTIONS EDITION 1

enwiabe

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Cthulhu

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dcc

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zzdfa

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dcc

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