Define random variable distribution given standard normal random variable

Question

I have a homework but I don't know how to solve it or what should I do. Kindly help me or guide me.

Let a constant $a$ satisfy that $\int_{0}^{a}{x^2 e^{-\frac{x^2}{2} }dx} = \int_{a}^{\infty}{x^2 e^{-\frac{x^2}{2} }dx}$

Suppose $X$ is a standard normal random variable. Define $Y$ as follows

$$ Y = \left\{ \begin{array}{ll} X & \quad if|X|\geq a \\ -X & \quad if|X| < a \end{array} \right. $$

(a) What is the distribution of $Y$?

(b) Show that $X$ and $Y$ are uncorrelated.

(c) Show that $X$ and $Y$ are not independent.

Actually I don't understand the purpose of given integration function, like how or what can I do with the function. What I understand so far (maybe true or false), I consider when $|X| \geq a$ that's mean $Y = \int_{a}^{\infty}{x^2 e^{-\frac{x^2}{2} }dx}$ and the opposite when $|X| < a$. If I consider this, that's mean Y is uniform distribution because nothing different when |X| is greater or less than $a$. Then to show X and Y are uncorrelated, $Cov(X,Y)=E[XY]-E[X]E[Y]=0$ that means $E[XY]=E[X]E[Y]$. But this understanding is the opposite of (c) question, since this apply when X and Y are independent. Thank you so much for your help.

Answer 1

(a) What happens when you multiply a symmetric distribution by -1? That is, what's the difference between the distribution of $X$ and $-X$?

(b) If two random variables are uncorrelated, then that implies that their covariance is 0. Covariance is $E[(X-E[X])(Y-E[Y])]$, which is made simpler by the fact that $E[X]=E[Y]=0$. What happens when you integrate that across the domain of the function? (The definition of $a$ should come in handy.)

(c) Independence implies that $P(X=x, Y=y)$ factorizes to $P(X=x)P(Y=y)$. Can you exhibit a pair $(x, y)$ where that doesn't hold?