A random valued input voltage (which is known to have a Gaussian distribution with mean = 3, and variance = 1) is amplified (i.e. multiplicative gain) using an amplifier that has two possible values of gain: {1, 5}, both equally likely. Further, the amplifier itself adds noise to its output (self noise) that is known to be a standard Gaussian. DenoteX

Xto be the random variable corresponding to the input voltage,G

Gto be the random variable corresponding to the amplifier gain,Y

Yto be the random variable corresponding to the amplifier output, andZ

Zto be the random variable corresponding to the self noise. Thus, we have:

Y=GX+Z where G, X, Z are mutually independent random variables

Determine Var(Y) and P(G=5|y=6)