Show that Constants are Normally Distributed

Let X1 and X2 denoted independent, normally distributed random variables, not necessarily the same meanor 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.

Please see the attached file.

Let X1 and X2 denote independent, normally distributed random variables, not necessarily having the same mean or variance. Show that, for any …

The expert shows the constants which are normally distributed.