There are two types of batteries in a bin. When in use, type i batteries last (in hours) an exponentially distributed time with rate

$$\lambda _ { i } , i = 1,2$$

. A battery that is randomly chosen from the bin will be a type i battery with probability

$$p _ { i } , \sum _ { i = 1 } ^ { 2 } p _ { i } = 1$$

If a randomly chosen battery is still operating after t hours of use, what is the probability that it will still be operating after an additional s hours?

Let

$$a _ { n } = H _ { n } - \ln n,$$

where

$$H_n$$

is the nth harmonic number:

$$H _ { n } = 1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \dots + \frac { 1 } { n }$$

(a) Show that

$$a _ { n } \geq 0$$

for

$$n \geq 1.$$

Hint: Show that

$$H _ { n } \geq \int _ { 1 } ^ { n + 1 } \frac { d x } { x }.$$

(b) Show that

$$\left\{ a _ { n } \right\}$$

is decreasing by interpreting

$$a _ { n } - a _ { n + 1 }$$

as an area. (c) Prove that

$$\lim _ { n \rightarrow \infty } a _ { n }$$

exists. This limit, denoted

$$\gamma,$$

is known as Euler's Constant. It appears in many areas of mathematics, including analysis and number theory, and has been calculated to more than 100 million decimal places, but it is still not known whether

$$\gamma$$

is an irrational number. The first 10 digits are

$$\gamma \approx 0.5772156649.$$

The annual rainfall in Cleveland, Ohio is approximately a normal random variable with mean 40.2 inches and standard deviation 8.4 inches. What is the probability that (a) next year’s rainfall will exceed 44 inches? (b) the yearly rainfalls in exactly 3 of the next 7 years will exceed 44 inches? Assume that if

$$A_i$$

is the event that the rainfall exceeds 44 inches in year i (from now), then the events

$$A_i$$

,

$$i \geq 1$$

, are independent.

Use a graphing utility to graph the equation. Use a standard viewing window. Approximate any x- or y-intercepts of the graph. $y-4 x=x^{2}(x-4)$

A stick of length 1 is broken at a uniformly random point, yielding two pieces. Let $X$ and $Y$ be the lengths of the shorter and longer pieces, respectively, and let $R = X/Y$ be the ratio of the lengths $X$ and $Y$ . (a) Find the CDF and PDF of $R$. (b) Find the expected value of $R$ (if it exists). (c) Find the expected value of $1/R$ (if it exists).

A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the probability that (a) you are winning after 34 bets; (b) you are winning after 1,000 bets; (c) you are winning after 100,000 bets. Assume that each roll of the roulette ball is equally likely to land on any of the 38 numbers.

There are n distinct types of coupons, and each coupon obtained is, independently of prior types collected, of type i with probability pi,

$$\sum _ { i = 1 } ^ { n } p _ { i } = 1$$

. (a) If n coupons are collected, what is the probability that one of each type is obtained? (b) Now suppose that

$$p _ { 1 } = p _ { 2 } = \cdots = p _ { n } = 1 / n$$

. Let

$$E_i$$

be the event that there are no type i coupons among the n collected. Apply the inclusion–exclusion identity for the probability of the union of events to

$$P \left( \bigcup _ { i } E _ { i } \right)$$

to prove the identity

$$n ! = \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( n - k ) ^ { n }$$

Find the slope of the line that passes through the points. (3, -11) and (0, 4)

Suppose your visit distant countries and experience the following temperatures in degrees Celsius. Change the temperatures to degrees Fahrenheit. $- 18 ^ { \circ } \mathrm { C }$

For any real number y, define

$$y ^ { + }$$

by

$$y ^ { + } = \begin{array} { l l } { y , } & { \text { if } y \geq 0 } \\ { 0 , } & { \text { if } y < 0 } \end{array}$$

Let c be a constant. (a) Show that

$$E \left[ ( Z - c ) ^ { + } \right] = \frac { 1 } { \sqrt { 2 \pi } } e ^ { - c ^ { 2 } / 2 } - c ( 1 - \Phi ( c ) )$$

when Z is a standard normal random variable. (b) Find

$$E \left[ ( X - c ) ^ { + } \right]$$

when X is normal with mean

$$\mu$$

and variance

$$\sigma ^ { 2 }$$

.

Balls are randomly removed from an urn that initially contains 20 red and 10 blue balls. (a) What is the probability that all of the red balls are removed before all of the blue ones have been removed? Now suppose that the urn initially contains 20 red, 10 blue, and 8 green balls. (b) Now what is the probability that all of the red balls are removed before all of the blue ones have been removed? (c) What is the probability that the colors are depleted in the order blue, red, green? (d) What is the probability that the group of blue balls is the first of the three groups to be removed?

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2, 4, 6, 8, ...} is specified by the function f(n)=2n, where n=1, 2, 3,... The limit of such a sequence is $\lim _ { n \rightarrow \infty } f ( n )$, provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. $\left\{ 0 , \frac { 1 } { 2 } , \frac { 2 } { 3 } , \frac { 3 } { 4 } , \dots \right\}$, which is defined by f(n)=$\frac { n - 1 } { n }$, for n=1, 2, 3, ...

If X is an exponential random variable parameter

$$\lambda = 1$$

, compute the probability density function of the random variable Y defined by Y=log X.

Find the LCD. List the multiples or use prime factorization. $\frac { 7 } { 21 } , \frac { 3 } { 9 },$ and $\frac { 5 } { 7 }$