Question

Let $\dpi{80} X_{1}\sim N(4.2,1)$ and $\dpi{80} X_{2}\sim N(12,70)$ be two Gaussian random variables.

(1) Sketch the PDFs of $\dpi{80} X_{1}$ , $\dpi{80} X_{2}$ on the same chart.

(2) Assuming $\dpi{80} X_{1}$ , $\dpi{80} X_{2}$ are independent, compute $\dpi{80} \mathbb{P}[X_{1}>X_{2}]$ .

Answer 1

code in R

par(mfrow = c(2,1))

x <- rnorm(1000, mean=4.2, sd=1)
hist(x, probability=TRUE)
xx <- seq(min(x), max(x), length=100)
lines(xx, dnorm(xx, mean=4.2, sd=1))

y <- rnorm(1000, mean=12, sd= sqrt(70))
hist(y, probability=TRUE)
yy <- seq(min(y), max(y), length=100)
lines(yy, dnorm(yy, mean=12, sd= sqrt(70)))

2)

x1 = N( 4.2 , 1)

X2 = N(12,70)

X1 - X2 = N(4.2 - 12 , 1 + 70)

= N(-7.8 , 71)

P(X1 > X2)

= P(X1 - X2 > 0)

= P(Z > (0 -(-7.8))/sqrt(71))

=P(Z > 0.92568 )

=

0.1773062