Question

et Xi and X2 be independent random variables with common distribution NORM(8,1), where θ is a real number. Let Y = X1 + X a) Find the p.d.f. of Y. 4. L 2. b) Use the appropriate table to find P(Y> 20 + 2)

Answer 1

4)

a)

We know that if $X\sim N(\mu,\sigma^{2})$ , then its PDF is given by

$f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-1}{2 \sigma^{2}}(x-\mu)^{2}},-\infty<X<\infty$

Now, $X_{i} \sim N(\theta,1)$

So, $Y=X_{1}+X_{2}\sim N(\theta+\theta,1+1)\sim N(2\theta,2)$

Thus PDF of Y is:

$g(y)=\frac{1}{\sqrt{2\pi}\sqrt{2}}e^{\frac{-1}{2 (2)}(x-2\theta)^{2}},-\infty<Y<\infty$

$g(y)=\frac{1}{\sqrt{2\pi}\sqrt{2}}e^{\frac{-1}{4}(x-2\theta)^{2}},-\infty<Y<\infty$

b)

Required probability = $P(Y>2\theta +2)=P(Z>\frac{(2\theta +2)-2\theta}{\sqrt{2}})=P(Z>1.41)=0.07927$