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Show that Constants are Normally Distributed

Let X1 and X2 denoted independent, normally distributed random variables, not necessarily the same mean or variance. Show that any constants "a" and "b", Y= aX1 + bX2 is normally distributed.

Help:
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I'm not sure if I can just prove it by showing what I have below.
where,
SX2 ---> INTEGRATION FROM X_2
SX1 ---> INTEGRATION FROM X_1

E[aX1 + bX2 ] = SX2 SX1 (aX1 + bX2)f(X1, X2) dX1 dX2
= a SX2 SX1 X1 f(X1, X2) dX1 dX2
+ b SX1 SX2 X2 f(X1,X2) dX1 dX2
= aE[X1] + bE[X2 ]
and then do Variance as well.

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