Discrete random variables and distributions_imported Flashcards
What are discrete random variables?
Discrete random variables are variables that can take on a finite or countably infinite number of distinct values. These values are determined by the outcome of a random experiment or process.
What kind of values can discrete random variables take?
Discrete random variables can take on specific, distinct values, usually represented by integers. For example, the number of heads in a series of coin tosses or the outcome of rolling a fair six-sided die.
How are probabilities associated with discrete random variables?
Each possible value of a discrete random variable is associated with a probability of occurrence. These probabilities are often represented using a probability mass function (PMF).
What does the probability distribution of a discrete random variable show?
The probability distribution of a discrete random variable shows the probabilities of all possible outcomes. It provides insights into the likelihood of each value occurring.
Can you give an example of a discrete random variable and its probability distribution?
Certainly! Consider a random variable X representing the number of heads in two coin tosses. It can take values 0, 1, or 2. The associated probabilities might be P(X = 0) = 0.25, P(X = 1) = 0.5, and P(X = 2) = 0.25.
What is the requirement for the sum of probabilities in a discrete probability distribution?
The sum of the probabilities for all possible values of a discrete random variable must equal 1. This ensures that one of the possible outcomes will indeed occur.
Are there any measures used to describe the distribution of discrete random variables?
Yes, discrete random variables have well-defined mean (expected value) and variance, which provide measures of central tendency and spread for the variable’s distribution.
Can you provide some common examples of discrete random variables?
Certainly! Examples of discrete random variables include the outcomes of dice rolls (e.g., rolling a six-sided die), the number of emails received in an hour, the number of defects in a production batch, and the number of people in a household.
What is a Bernoulli random variable?
A Bernoulli random variable represents a binary outcome (success or failure) of a single trial.
Example: What is the probability of getting a Heads when flipping a fair coin?
The probability of getting a Heads when flipping a fair coin is 0.5.
What is a Binomial random variable?
A Binomial random variable represents the number of successes in a fixed number of independent Bernoulli trials.
Example: What is the probability of getting exactly 3 Heads in 5 coin tosses?
The probability of getting exactly 3 Heads in 5 coin tosses is given by the binomial distribution formula.
What is a Poisson random variable?
A Poisson random variable represents the number of events occurring in a fixed interval of time or space.
Example: What is the probability of observing 2 car accidents at a specific intersection in a given hour?
The probability of observing 2 car accidents in a given hour can be calculated using the Poisson distribution formula.
What is a Geometric random variable?
A Geometric random variable represents the number of trials needed for the first success in a sequence of Bernoulli trials.
Example: How many times do you need to roll a die to get the first 6?
The expected number of rolls needed to get the first 6 on a fair 6-sided die is 6.
What is a Hypergeometric random variable?
A Hypergeometric random variable represents the number of successes in a sample drawn without replacement from a finite population.
Example: If you draw 2 cards from a deck without replacement, what’s the probability of getting exactly 1 Ace?
The probability of drawing exactly 1 Ace from a deck when drawing 2 cards without replacement can be calculated using the hypergeometric distribution formula.
What is a Negative Binomial random variable?
A Negative Binomial random variable represents the number of trials needed for a fixed number of successes in a sequence of Bernoulli trials.
Example: How many times do you need to flip a coin to get 3 Heads?
The expected number of coin flips needed to get 3 Heads in a row can be calculated using the negative binomial distribution formula.
What is a Uniform random variable?
A Uniform random variable represents outcomes that are equally likely within a certain range.
Example: What is the probability of rolling a 4 on a fair 6-sided die?
The probability of rolling a 4 on a fair 6-sided die is 1/6.
What is a Discrete Uniform random variable?
A Discrete Uniform random variable represents outcomes that are equally likely among a finite set of values.
Example: What is the probability of rolling a 2 on a fair 4-sided die?
The probability of rolling a 2 on a fair 4-sided die is 1/4.
What is a Categorical random variable?
A Categorical random variable represents outcomes from a category or label.
Example: What is the probability of selecting a red candy from a bag containing red, green, and blue candies?
The probability of selecting a red candy from the bag can be determined based on the ratio of red candies to the total number of candies.
Types of discrete probability distributions
Bernoulli Distribution, Binomial Distribution, Poisson Distribution, Geometric Distribution, Hypergeometric Distribution, Negative Binomial Distribution, Discrete Uniform Distribution, Categorical Distribution, Multinomial Distribution.
What is a Bernoulli Distribution?
A Bernoulli distribution represents a single trial with two possible outcomes – success (usually denoted as 1) or failure (usually denoted as 0).
Example: What is the probability of flipping a coin and getting heads?
The probability of flipping a coin and getting heads is 0.5 in a fair coin toss.
What is a Binomial Distribution?
The binomial distribution represents the number of successes in a fixed number of independent Bernoulli trials.
Example: If you roll a fair six-sided die five times, what’s the probability of getting exactly two fives?
The probability of getting exactly two fives in five rolls of a fair six-sided die can be calculated using the binomial distribution formula.
What is a Poisson Distribution?
The Poisson distribution models the number of events occurring in a fixed interval of time or space when events happen randomly and independently.
Example: In a busy restaurant, what is the probability of receiving exactly three customer complaints in an hour?
The probability of receiving exactly three customer complaints in an hour can be calculated using the Poisson distribution formula.
What is a Geometric Distribution?
The geometric distribution represents the number of trials needed for the first success in a sequence of independent Bernoulli trials.
Example: How many times do you need to roll a die to get the first six?
The expected number of rolls needed to get the first six on a fair six-sided die can be calculated using the geometric distribution formula.
What is a Hypergeometric Distribution?
The hypergeometric distribution represents the number of successes in a sample drawn without replacement from a finite population.
Example: In a deck of cards, if you draw five cards without replacement, what’s the probability of getting exactly two Aces?
The probability of getting exactly two Aces when drawing five cards from a deck without replacement can be calculated using the hypergeometric distribution formula.
What is a Negative Binomial Distribution?
The negative binomial distribution represents the number of trials needed to achieve a fixed number of successes in a sequence of independent Bernoulli trials.
Example: How many times do you need to flip a coin to get three heads?
The expected number of coin flips needed to get three heads in a row can be calculated using the negative binomial distribution formula.
What is a Discrete Uniform Distribution?
The discrete uniform distribution represents outcomes that are equally likely among a finite set of values.
Example: If you roll a fair six-sided die, what’s the probability of getting a three?
The probability of rolling a three on a fair six-sided die is 1/6.
What is a Categorical Distribution?
The categorical distribution represents outcomes from a category or label.
Example: In a survey, what is the probability of someone choosing “Yes” from the options “Yes,” “No,” and “Maybe”?
The probability of someone choosing “Yes” from the options can be determined based on the categorical distribution probabilities.
What is a Multinomial Distribution?
The multinomial distribution represents the probabilities of outcomes in a categorical experiment with more than two categories.
Example: In a bag of colored marbles, what’s the probability of drawing two red, three blue, and one green marble in six draws?
The probability of drawing the specified combination of marbles can be calculated using the multinomial distribution formula.
