What is the p.d.f. of the sum of two normal distributions?

[thesandbox]

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.

µ_Y=E[Y]=E[X₁+X₂]=E[X₁]+E[X₂]=µ₁+µ₂

σ_Y²=E[Y²] - µ_Y²

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.

 

[vela]

It's kind of hard to make suggestions without knowing more about what you know. What do you know about sums of random variables?

 

[thesandbox]

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.

µ_Y=E[Y]=E[X₁+X₂]=E[X₁]+E[X₂]=µ₁+µ₂

σ_Y²=E[Y²] - µ_Y²

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.

Figured it out. The above doesn't account for all moments (rth)

M_{X1 + X2} = M_X1(t)⋅M_X2(t)

M_Xi=e^{\mu_it + (1/2)σ_i2t2}

Following N~(\mu₁ + \mu₂ , σ₁² + σ₂²)