Answer Happy

21 3. In statistics, the standard normal distribution f(x) = -e plays an important role. The area under this curve between c = a and x = b gives the probability that normally-distributed random variable has a z-score between a and b. As this is a probability density function, the total area under the curve must be equal to 1. In this question we will verify that this is the 1 case. I.e. we will compute dx e e . How are (a) Can you find an antiderivative for f(x)? Either give the antiderivative or explain in a sentence or two why you are unable to find one. 1 (b) Consider the function g(y) = ✓27 1 (c) If we let I = * dx, and we let h(x, y) = f(x)g(y), show that Where R is the entire x, y-plane. (d) Change to polar coordinates and write h(r, ) (change variables to write h in terms of r [ f(x)dt and S*g(u)dy related? I SA P(,y)dA = P. and ). (e) Compute on the right) S S M(x, y)dA= S S M(,C)dA (You should be able to evaluate the integral 1 (f) Show that I = 1, in other words, show that dx=1 V2T - 1 (g) Explain why this technique cannot be used to evaluate are finite real numbers. Ś dx, where a and b 2п