0

I'm reading through the solutions of my book and they skip over a lot of steps so can someone please explain where their answers came from? I haven't done probability in a while so if there are some basic principles or theorems I'm forgetting I wouldn't be surprised.

1) I'm given that $X$ is a standard normal RV. The book says $E[e^{uX+vX}] = e ^{(u+v)^2/2}$. Where did that come from? I thought that I would need to do $\int_{-\infty}^{\infty}e^{-.5x^2}e^{(u+v)x}dx$, which gives a different answer.

2) Not expected value per se, but I have $\int_{x-a/2}^{x+a/2}e^{-.5x^2}dx$ and again the book just jumps to the answer that its $(a)e^{-x^2/2}$. Again, how did they get there?

user559412
  • 115
  • 7
  • (1) you are correct. Indeed you would have to perform $\int_{-\infty}^{\infty}e^{-.5x^2}e^{(u+v)x}dx$

    .You've got a mistake in (2) .. How can the boundaries of your integral depend on $x $ ?

     – Ahmad Bazzi Sep 10 '18 at 03:38

1 Answers1

2

For 1), looking at how to compute the MGF of the standard normal distribution may help. Note that your integral is not quite right (you forgot the constant $(2\pi)^{-1/2}$).

For 2) the integral does not make sense because a) you are using $x$ for two different meanings. Once you fix this, the "answer" is not exact, but an approximation using the fundamental theorem of calculus.

angryavian
  • 89,882