Question

Suppose that X ∼ N(2,1) and Y ∼ N(3,2). Assuming X and Y are independent what is the distribution of X + Y ? *

N(3,5)

N(5,3)

N(3,3)

N(5,5)

​

Answer 1

Answer:

Hence answer is B. $X+Y \sim N(5,3)$

Step-by-step explanation:

Step 1: Distributions are objects that broaden the traditional idea of functions in mathematical analysis. They are often referred to as Schwartz distributions or generalised functions. Functions with no classically defined derivatives can be distinguished using distributions.

Step 2: The highest point on the curve represents the most typical or modal value, which is typically quite near to the population's average (mean). The Maxwell-Boltzmann distribution law, which determines the likelihood that a gas molecule would be discovered with velocity components u, v, and w in the x, y, and z directions, is a well-known example from physics. One can add as many variables as they like in a distribution function.

Step 3: Here $X \sim N(2,1)$ and $Y \sim N(3,2)$

Now here it is given that X and $\mathrm{Y}$ are independent

So, $E(X+Y)=E(X)+E(Y)=2+3=5$

Now $V(X+Y)=V(X)+V(Y)+2 \{Cov}(X, Y)$

As $\mathrm{X}$ and $\mathrm{Y}$ are independent ${Cov}(X, Y)=0$

So $V(X+Y)=V(X)+V(Y)=1+2=3$

Hence answer is B.$X+Y \sim N(5,3)$

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