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Mathematics Question Thread
fun_jirachi:
Welcome to the forums!
They're basically asking you to take the reciprocals then rationalise the denominator without explicitly telling you the second step.
This is the true statement you're supposed to find. Note that you can't actually present this as a proof because you're starting with what you're trying to prove, but working backwards from the last line of working is an acceptable proof of the inequality in the question (and this method of 'discovering a proof' is very handy in a lot of HS maths, especially compared to blindly using brute force). Working backwards is fine because you're starting with a statement of fact, then proving (or maybe in some other case disproving) the statement presented in the question.
Hope this helps
Maroon and Gold Never Fold:
Attached a question I am unable to do.
RuiAce:
--- Quote from: Maroon and Gold Never Fold on May 31, 2021, 11:18:56 pm ---Attached a question I am unable to do.
--- End quote ---
For \(y>0\),
\begin{align*}
F_Y(y) &= P(Y\leq y)\\
&= P(\sqrt{X} \leq y)\\
&= P(X \leq y^2) \tag{square both sides}\\
&= F_X(y^2) \tag{because we now have $X$}\\
&= 1-e^{-y^2}.
\end{align*}
You can then find the density function through the usual way (taking derivative).
For any future questions, you should post your attempts at solving them, no matter how right or wrong they are. That way we can be sure you've given them a go first, and can also understand your thought process to give you actual valuable help.
Maroon and Gold Never Fold:
--- Quote from: RuiAce on June 01, 2021, 12:17:31 am ---
For any future questions, you should post your attempts at solving them, no matter how right or wrong they are. That way we can be sure you've given them a go first, and can also understand your thought process to give you actual valuable help.
--- End quote ---
Thanks for your help. I was really stuck on the wording of the question and I'm still a bit lost on what I'm sort of meant to do.
RuiAce:
--- Quote from: Maroon and Gold Never Fold on June 01, 2021, 08:10:45 am ---Thanks for your help. I was really stuck on the wording of the question and I'm still a bit lost on what I'm sort of meant to do.
--- End quote ---
You started with a random variable \(X\). You were given the CDF of \(X\).
You then pull out a new random variable \(Y\). You are told that this new random variable is related to the old one, by the equation \( Y = \sqrt{X}\). (So you see that \(Y\) is in fact a function of another random variable, and hence also a random variable.)
Since \(Y\) is a new random variable, it should also have a CDF. You are asked to find this CDF in the first part. (And then using it, you can find the PDF in the second part.)
(But how would you be able to find the CDF of \(Y\) to begin with? Well, you're given the relationship \( Y = \sqrt{X}\), and the CDF of \(X\). So somehow you had to puzzle the two together.)
The actual technique used is understandably one you might've not seen before. But as for what the question was asking, you needed to realise most of the above.
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