Discrete Random Variables: 동영상 및 연습문제

Each of the following scenarios can be described as a discrete random variable, a continuous random variable, or not a random variable. Which scenarios are continuous random variables?

a. The number of pages in various novels in a library

b. The amount of milk (in liters) a cow produces in a day

c. The number of languages spoken by university professors

d. Types of plants in a botanical garden

e. The time (in minutes) it takes for a student to complete a math quiz

During a health survey, people are arranged in groups of $5$ and asked if they exercise regularly. The random variable $x$ denotes the number of people in each group who say yes. The given probabilities for $x=0,1,2,3,4,5$ are $0.05,0.15,0.30,0.30,0.15,0.05$ respectively. Confirm if this is a probability distribution and find the mean and standard deviation.

In a study on the effectiveness of different flu vaccines, four vaccines were tested with the following probabilities of preventing the flu:

Is this a probability distribution, and if so, what are its mean and standard deviation?

A local bakery is analyzing the number of cakes sold per day. The random variable $x$ represents the number of cakes sold. The probabilities are as follows:

Is this a probability distribution? If so, find its mean and standard deviation.

A health magazine conducted a poll among $800$ readers about their weekly exercise habits. The question was, 'Do you exercise more than $3$ times a week?' The responses were collected as shown below.

Determine if the table represents a probability distribution, and if so, find its mean and standard deviation.

Given the mean number of books published by an indie publisher is $4$ per month, use the Poisson distribution to find the probability that they will publish exactly $6$ books in a particular month.

Given that the mean number of solar flares observed from a particular satellite is $2.5$ per month, find the expected number of months with at least one solar flare in a $24$-month observation period. The probability distribution follows the Poisson model.

Assuming the average number of customers visiting a small cafe is $20$ per day, calculate the probability that exactly $15$ customers will visit the cafe on a given day. The probability distribution follows the Poisson model.

With an average of $2$ volcanic eruptions per year in a certain area, use the Poisson distribution to find the expected number of years with exactly $3$ eruptions in a $100$-year period.

A game has a probability distribution with $P\left(z\right)=0.25$ for $z$ values of $1,2,3,$ and $4$. Calculate the mean of the random variable $z$.

A game has a probability distribution with $P\left(z\right)=0.25$ for $z$ values of $1,2,3,$ and $4$. Is the mean value of the random variable $z$ a statistic or a parameter?

A hospital reports an average of $365$ patients admitted for treatment each year. Determine the mean number of patients admitted per day.

In a small city, an average of $15$ cars are reported stolen each month. The probability distribution follows the Poisson model. What is the probability that on a given day, no cars are reported stolen?

In a small city, an average of $15$ cars are reported stolen each month. The probability distribution follows the Poisson model. What is the probability that on a given day, more than one car is stolen?

During a particular month, a hospital recorded a total of $120$ births. Assuming that births occur randomly and independently at a constant average rate, use the Poisson distribution to determine the probability that on a randomly selected day, exactly $3$ births occur. Assume that $1$ month is equal to $30$ days.

A local coffee shop serves an average of $90$ customers per hour. Using the Poisson distribution, determine the probability that exactly $10$ customers will be served in the next $10$ minutes.

In a particular year, a large metropolitan hospital recorded a total of $7300$ emergency room visits. The probability distribution follows the Poisson model. Calculate the probability that on a given day, there are exactly $20$ emergency room visits.

In a small town library, there were $4380$ books loaned out over the course of $365$ days. Calculate the average number of books loaned out per day.

A survey of office workers recorded the number of password reset requests made by each employee in a month. The data below represents the probability distribution for the number of reset requests:

Is the random variable for the number of reset requests discrete or continuous? Explain.

With an average of $2$ volcanic eruptions per year in a certain area, use the Poisson distribution to find the expected number of years with exactly $3$ eruptions in a $100$-year period. How does this result compare to a recent $100$-year period in which there were $6$ years with exactly $3$ eruptions? Does the Poisson distribution appear to model the situation well?

In a quality control test, a random variable represents the number of defective items found in a batch of $20$ products. What is the random variable, what are its possible values, and are its values numerical?

Let $x$ represent the number of books checked out from a library by a student in one visit. Is $x$ a discrete or continuous random variable? Choose the best answer.

A factory finds that $1.2\%$ of its products are defective. Suppose the products are inspected one at a time.

(a) Compute the mean, variance, and standard deviation for the number of inspections needed to find the first defective product.

(b) What is the expected number of inspections before finding a defective item?

A basketball player has a $0.72$ probability of making a free throw. What is the probability that her first successful free throw occurs on her second attempt?

A basketball player has a free-throw success rate of $0.12$. What is the probability that her first successful free throw occurs on her $6$^th attempt?

A website gets an average of $2$ hits per second. What is the probability that it gets exactly $5$ hits in a particular second? Use the Poisson distribution.

The number of smartphones owned per student in a college dormitory is recorded as follows:

Calculate the mean, variance, and standard deviation of this probability distribution.

How does the interpretation of the variable $x$ differ between the binomial and Poisson distributions?

The average number of customer support calls received by a tech company per hour is $4.3$. Find the probability that the number of calls received in a randomly selected hour is:

i. Exactly $0$

ii. Exactly $4$

iii. Exactly $7$

A customer service agent successfully resolves a caller’s issue on the first attempt $72\%$ of the time. Find the probability that:

i. The first successful resolution occurs on the third call,

ii. The first successful resolution occurs on or before the second call, and

iii. The agent does not successfully resolve any of the first three calls.