Question: The probability density function (PDF) of a sum of two independent continuous random variablesXandYis given by the convolution of the PDFs, fXandfY: fX+Y(z) = fX(x)fY(zx)dx.

The probability density function (PDF) of a sum of two independent continuous random variablesXandYis given by the convolution of the PDFs,

fXandfY:

fX+Y(z) =

fX(x)fY(zx)dx.

Show that a sum of two independent standard normal variables results in a normal variable. Find the PDF of such a sum. Give your solutions in two ways: 1) by using the convolution function above; 2) by using moment generating functions.

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