Given the ID number 133477566 and the digit frequency probability mass function below denoted as P_D(d):

P_D(d) =

{1/9, if d = 1,4,5}

{2/9, if d = 3,7,6}

{0, otherwise}

1) Let X = S₁ + S₂ equal the value of the summation of two different independent samples from P_D(d). Determine the sample space for X in which all of the values that have been list have probabilities that are non zero values.

2) Determine the probability mass function P_X[x] for the variable X which is a random variable.

3) Determine the value of X that is expected, and provide proof in the answer.