Prove by hand the following result if XN225 and YN31 are ind

Prove \"by hand\" the following result: if X~N(2,25) and Y~N(3,1) are independent then X+Y~N(5,26). \"By hand\" means that you can not use moment generating functions.

Solution

we just need to see if the sum of X+Y for the mean and sigma is the same as they said

mean of X + Y = 2 + 3 = 5

SD of X + Y = 25 + 1 = 26

so we prove that X+Y follow a normal distribution with mean = 5 and SD = 26