Theory and ExamplesNormal probability distribution The

Chapter 8, Problem 79E

(choose chapter or problem)

Problem 79E

Theory and Examples

Normal probability distribution The function

is called the normal probability density function with mean μ and standard deviation σ. The number μ tells where the distribution is centered, and σ measures the “scatter” around the mean.

From the theory of probability, it is known that

In what follows, let μ = 0 and σ = 1

a. Draw the graph of ƒ. Find the intervals on which ƒ is increasing, the intervals on which ƒ is decreasing, and any local extreme values and where they occur.

b. Evaluate

for n = 1, 2, and 3.

c. Give a convincing argument that

(Hint: Show that for and for 0 < f(x) < e–x/2 for x > 1, and for b> 1

)