91Ó°ÊÓ

The time(in minutes) until the next bus departs a major bus depot follows a distribution with \(f(x)=\frac{1}{20}\) where \(x\) goes from \(25\) to \(45\) minutes.

a. Define the random variable. \(X=\).

b. \(X~\)

c. Graph the probability distribution.

The time (in minutes) until the next bus departs a major bus depot follows a distribution with $f\left(x\right)=\frac{1}{20}$ where $X$ goes from 25 to 45 minutes.

a. Define the random variable. $X=$

b. $X~$

c. Graph the probability distribution.

d. The distribution is (name of distribution). It is (discrete or continuous).

e. $\mathrm{μ}=$

f. $\mathrm{σ}=$

g. Find the probability that the time is at most 30 minutes. Sketch and label a graph of the distribution. Shade the area of interest. Write the answer in a probability statement.

h. Find the probability that the time is between 30 and 40 minutes. Sketch and label a graph of the distribution. Shade the area of interest. Write the answer in a probability statement.

i. $P(25<x<55)=$ . State this in a probability statement, similarly to parts g and h, draw the picture, and find the probability.

j. Find the ${90}^{\text{th}}$ percentile. This means that $90\%$ of the time, the time is less than

k. Find the ${75}^{\text{th}}$ percentile. In a complete sentence, state what this means. (See part j.)

1. Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes.

Suppose that the length of long distance phone calls, measured in minutes, is known to have an exponential distribution with the average length of a call equal to eight minutes.

a. Define the random variable. $X=$

b. Is X continuous or discrete?

c. $X~$

d. $\mathrm{μ}=$

e. $\mathrm{σ}=$

f. Draw a graph of the probability distribution. Label the axes.

g. Find the probability that a phone call lasts less than nine minutes.

h. Find the probability that a phone call lasts more than nine minutes.

i. Find the probability that a phone call lasts between seven and nine minutes.

j. If 25 phone calls are made one after another, on average, what would you expect the total to be? Why?

Suppose that the useful life of a particular car battery, measured in months, decays with parameter 0.025. We are interested in the life of the battery.

a. Define the random variable. X = _________________________________.

b. Is X continuous or discrete?

c. X ~ ________

d. On average, how long would you expect one car battery to last?

e. On average, how long would you expect nine car batteries to last, if they are used one after another?

f. Find the probability that a car battery lasts more than 36 months.

g. Seventy percent of the batteries last at least how long?

The percent of persons (ages five and older) in each state who speak a language at home other than English is approximately exponentially distributed with a mean of $9.848$. Suppose we randomly pick a state.

a. Define the random variable.$X=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

b. Is $X$continuous or discrete?

c.$X~\_\_\_\_\_\_\_\_$

d.$\mathrm{μ}=\_\_\_\_\_\_\_\_$

e.$\mathrm{σ}=\_\_\_\_\_\_\_\_$

f. Draw a graph of the probability distribution. Label the axes.

g. Find the probability that the percent is less than $12$.

h. Find the probability that the percent is between eight and $14$.

i. The percent of all individuals living in the United States who speak a language at home other than English is $13.8$.

i. Why is this number different from $9.848\%$?

ii. What would make this number higher than $9.848\%$?

The time (in years) after reaching age $60$that it takes an individual to retire is approximately exponentially distributed with a mean of about five years. Suppose we randomly pick one retired individual. We are interested in the time after age

$60$to retirement.

a. Define the random variable.$X=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

b. Is $X$continuous or discrete?

c.$X~=\_\_\_\_\_\_\_\_$

d.$\mathrm{μ}=\_\_\_\_\_\_\_\_$

e.$\mathrm{σ}=\_\_\_\_\_\_\_\_$

f. Draw a graph of the probability distribution. Label the axes.

g. Find the probability that the person retired after age $70$.

h. Do more people retire before age $65$or after age $65$?

i. In a room of $1,000$people over age $80$, how many do you expect will NOT have retired yet?

The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of $\backslash (150$.

a. Define the random variable.$X=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

b.$X~=\_\_\_\_\_\_\_\_$

c.$\mathrm{μ}=\_\_\_\_\_\_\_\_$

d.$\mathrm{σ}=\_\_\_\_\_\_\_\_$

e. Draw a graph of the probability distribution. Label the axes.

f. Find the probability that a car required over $\backslash )300$for maintenance during its first year.

What is the probability that a phone will fail within two years of the date of purchase?

a.$0.8647$

b. $0.4866$

c. $0.2212$

d. $0.9997$

What is the median lifetime of these phones (in years)?

a. $0.1941$

b. $1.3863$

c. $2.0794$

d. $5.5452$

Suppose that the longevity of a light bulb is exponential with a mean lifetime of eight years.

a. Find the probability that a light bulb lasts less than one year.

b. Find the probability that a light bulb lasts between six and ten years.

c. Seventy percent of all light bulbs last at least how long?

d. A company decides to offer a warranty to give refunds to light bulbs whose lifetime is among the lowest two percent of all bulbs. To the nearest month, what should be the cutoff lifetime for the warranty to take place?

e. If a light bulb has lasted seven years, what is the probability that it falls within the $8^{\text{th}}$ year.