The probability density function (p.d.f.) of the sum of two independent normal distributions, X1 ∼ N(μ1, σ1²) and X2 ∼ N(μ2, σ2²), results in another normal distribution Y = X1 + X2. The mean of Y is calculated as μY = μ1 + μ2, while the variance is σY² = σ1² + σ2². This conclusion is derived from the moment-generating functions (MGFs) of the individual distributions, which combine multiplicatively. Therefore, Y follows the distribution N(μ1 + μ2, σ1² + σ2²).
PREREQUISITESStudents in statistics, data scientists, and anyone involved in probability theory or statistical analysis who needs to understand the behavior of sums of normal distributions.
thesandbox said:Homework Statement
X1 ∼ N(μ1,σ12) and X2 ∼ N(μ2,σ22)
Let Y = X1 + X2
Find the p.d.f. of Y & label the distribution.
Homework Equations
The Attempt at a Solution
Not quite sure how to go about this problem.
µY=E[Y]=E[X1+X2]=E[X1]+E[X2]=µ1+µ2
σY2=E[Y2] - µY2
E[X12+ 2X1X2 +X22] - (µ1+µ2)2
I think that might be how you start it. Feel free to correct me and start over or continue. Thanks.