Lecture 19

Faculty of Science; Department of Mathematics
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Readings For This Lecture

Keshet Course Notes:

Chapter 8, pages 161 to 171 (up to subsection 8.5)

Summary

Group 13: Add a summary of the lecture in this space. Include examples, discussion, and links to external sources, if desired.

Exercises

1. Ants are always found within 1 mile of their hill. The fraction of ants found within a distance $x$ of their hill (where $0<x<1$ mile) is given by the function $D(x)=(4/\pi )arctan(x)$ . What is the median distance that ants are found from their hill? What is the most likely position to find an ant? What is the average or mean distance?

2. Consider the function $f(x)=\cos(\pi x/6)$ for $0<x<3$ . This function is asymmetric on the given interval. Find the corresponding (normalized) probability density. Find the most probable value of x. Compute the mean and the median.

3. Suppose that x is a variable that takes on random non-negative values only. Show that the function below is a probability density distribution. $f(x)={\frac {2}{(1+x)^{3}}},0<x<\infty$ .

Find the probability that $x$ takes on values in the interval $[1,4]$ .