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Hi all i have this question in which i don't really understand the line of reasoning.

Given rvs X ∼ N(0, 1), Y ∼ N(0, 4), is P(X > 3) < P(Y < –6)?

The reasoning is :

P(Y < –6) = P(Y > 6) < P(Y / 2 > 3) = P(X > 3).

I don't understand this part: P(Y > 6) < P(Y / 2 > 3)

Isn't P(Y / 2 > 3) exactly equal to P(Y > 6)?

Also i don't really understand why P(Y / 2 > 3) = P(X > 3)

Could anyone enlighten me?

Thanks :)

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1 Answer

The random variable $Y$ has a mean of $0$ and a standard deviation of $\sqrt4=2$, so $Y/2$ has a mean of $0$ and a standard deviation of $2/2=1$ and is therefore $\mathcal{N}(0,1)$, just like $X$. Thus, $$\Bbb P\left(\frac{Y}2>3\right)=\Bbb P(X>3)\;.$$

For the rest, I agree with you that $Y/2>3$ exactly when $Y>6$, so that $$\Bbb P\left(\frac{Y}2>3\right)=\Bbb P(Y>6)\;.$$ It follows that $\Bbb P(Y<-6)=\Bbb P(X>3)$. It appears to me that the $<$ that’s bothering you is probably just a misprint.

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