Finding distribution of X

Question

We have X=Z if U<=0.5 and X=-Z if U>0.5 where Z is a standard normal variable and U is a uniform random variable (0,1).

I want to find the distribution of X. I am mainly unsure of how to approach this because of the inclusion of the uniform r.v.

Noting that Z and U are independent.

I know that Fu(u) = u for 0<u<1.

Am I correct in stating the following:

Fx(x) = P(X<=x) = P(Z<=x) = Then it would be the integral from 0 to 0.5 of the standard normal pdf?

Fx(x) = P(X<=x) = P(-Z<=x)= P(Z>=-x) = 1-P(Z<=-x) = 1 - (negative integral from 0.5 to 1 of the standard normal pdf)

Any feedback would be appreciated.

Answer 1

$$P(X \leq x)=P(X \leq x, U\leq \frac1 2)+P(X \leq x, U> \frac1 2)$$ $$=P(Z \leq x, U\leq \frac1 2)+P(-Z \leq x, U> \frac1 2).$$ By independence this becomes $P(Z \leq x) P( U\leq \frac1 2)+P(-Z \leq x) P(U>\frac1 2)$ Can you continue?