Random variable defined as A with 50% chance and B with 50% chance

Note: this is a homework problem so please don’t give me the whole answer!

I have two variables, A and B, with normal distributions (means and variances are known). Suppose C is defined as A with 50% chance and B with 50% chance. How would I go about proving whether C is also normally distributed, and if so, what its mean and variance are?

I’m not sure how to combine the PDFs of A and B this way, but ideally if someone can point me in the right direction, my plan of attack is to derive the PDF of C and show whether it is or isn’t a variation of the normal PDF.

Hopefully it’s clear to you that C isn’t guaranteed to be normal. However, part of your question was how to write down its PDF.
@BallpointBen gave you a hint. If that’s not enough, here are some more spoilers…

Note that C can be written as:
$$C = T \cdot A + (1-T) \cdot B$$
for a Bernoulli random $T$ with $P(T=0)=P(T=1)=1/2$ with $T$ independent of $(A,B)$. This is more or less the standard mathematical translation of the English statement “C is A with 50% chance and B with 50% chance”.

Now, determining the PDF of C directly from this seems hard, but you can make progress by writing down the distribution function $F_C$ of C. You can partition the event $C \leq X$ into two subevents (depending on the value of $T$) to write:

$$F_C(x) = P(C \leq x) = P(T = 0 \text{ and } C \leq x) + P(T = 1\text{ and C }\leq x)$$

and note that by the definition of C and the independence of T and B, you have:

$$P(T=0\text{ and }C \leq x) = P(T=0\text{ and }B\leq x) = \frac12P(B\leq x) = \frac12 F_B(x)$$

You should be able to use a similar result in the $T=1$ case to write $F_C$ in terms of $F_A$ and $F_B$. To get the PDF of C, just differentiate $F_C$ with respect to x.