Chapter 2: Continuous Probability DensitiesContent check — Probability (Grinstead and Snell), Chapter 2: Continuous Probability Densities

Suppose you choose at random a real number $X$ from the interval $[\mathrm{2,10}]$⁠.

(a) Find the density function $f(x)$ and the probability of an event $E$ for this experiment, where $E$ is a subinterval $[a,b]$ of $[\mathrm{2,10}]$⁠.

(b) From (a), find the probability that $X>5$⁠, that $5<X<7$⁠, and that $X^2-12X+35>0$⁠.

Same as Exercise 2, but suppose $$f(x)=\frac{C}{x}.$$

Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density $$f(t)=\lambda e^{-\lambda t},$$ where $\lambda =1$⁠, so that the probability $P(0,T)$ that a particle will appear in the next $T$ seconds is $P([0,T])={\int }_0^T\lambda e^{-\lambda t}dt$⁠. Find the probability that a particle (not necessarily the first) will appear

(a) within the next second.

(b) within the next 3 seconds.

(c) between 3 and 4 seconds from now.

(d) after 4 seconds from now.

Choose a number $B$ at random from the interval $[\mathrm{0,1}]$ with uniform density. Find the probability that

(a) $1/3<B<2/3$⁠.

(b) $|B-1/2|\le 1/4$⁠.

(c) $B<1/4$ or $1-B<1/4$⁠.

(d) $3B^2<B$⁠.

Take a stick of unit length and break it into two pieces, choosing the break point at random. Now break the longer of the two pieces at a random point. What is the probability that the three pieces can be used to form a triangle?

At the Tunbridge World’s Fair, a coin toss game works as follows. Quarters are tossed onto a checkerboard. The management keeps all the quarters, but for each quarter landing entirely within one square of the checkerboard the management pays a dollar. Assume that the edge of each square is twice the diameter of a quarter, and that the outcomes are described by coordinates chosen at random. Is this a fair game?