# Discrete random variables and distributions_imported Flashcards

1
Q

What are discrete random variables?

A

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.

2
Q

What kind of values can discrete random variables take?

A

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.

3
Q

How are probabilities associated with discrete random variables?

A

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).

4
Q

What does the probability distribution of a discrete random variable show?

A

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.

5
Q

Can you give an example of a discrete random variable and its probability distribution?

A

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.

6
Q

What is the requirement for the sum of probabilities in a discrete probability distribution?

A

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.

7
Q

Are there any measures used to describe the distribution of discrete random variables?

A

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.

8
Q

Can you provide some common examples of discrete random variables?

A

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.

9
Q

What is a Bernoulli random variable?

A

A Bernoulli random variable represents a binary outcome (success or failure) of a single trial.

10
Q

Example: What is the probability of getting a Heads when flipping a fair coin?

A

The probability of getting a Heads when flipping a fair coin is 0.5.

11
Q

What is a Binomial random variable?

A

A Binomial random variable represents the number of successes in a fixed number of independent Bernoulli trials.

12
Q

Example: What is the probability of getting exactly 3 Heads in 5 coin tosses?

A

The probability of getting exactly 3 Heads in 5 coin tosses is given by the binomial distribution formula.

13
Q

What is a Poisson random variable?

A

A Poisson random variable represents the number of events occurring in a fixed interval of time or space.

14
Q

Example: What is the probability of observing 2 car accidents at a specific intersection in a given hour?

A

The probability of observing 2 car accidents in a given hour can be calculated using the Poisson distribution formula.

15
Q

What is a Geometric random variable?

A

A Geometric random variable represents the number of trials needed for the first success in a sequence of Bernoulli trials.

16
Q

Example: How many times do you need to roll a die to get the first 6?

A

The expected number of rolls needed to get the first 6 on a fair 6-sided die is 6.

17
Q

What is a Hypergeometric random variable?

A

A Hypergeometric random variable represents the number of successes in a sample drawn without replacement from a finite population.

18
Q

Example: If you draw 2 cards from a deck without replacement, what’s the probability of getting exactly 1 Ace?

A

The probability of drawing exactly 1 Ace from a deck when drawing 2 cards without replacement can be calculated using the hypergeometric distribution formula.

19
Q

What is a Negative Binomial random variable?

A

A Negative Binomial random variable represents the number of trials needed for a fixed number of successes in a sequence of Bernoulli trials.

20
Q

Example: How many times do you need to flip a coin to get 3 Heads?

A

The expected number of coin flips needed to get 3 Heads in a row can be calculated using the negative binomial distribution formula.

21
Q

What is a Uniform random variable?

A

A Uniform random variable represents outcomes that are equally likely within a certain range.

22
Q

Example: What is the probability of rolling a 4 on a fair 6-sided die?

A

The probability of rolling a 4 on a fair 6-sided die is 1/6.

23
Q

What is a Discrete Uniform random variable?

A

A Discrete Uniform random variable represents outcomes that are equally likely among a finite set of values.

24
Q

Example: What is the probability of rolling a 2 on a fair 4-sided die?

A

The probability of rolling a 2 on a fair 4-sided die is 1/4.

25
Q

What is a Categorical random variable?

A

A Categorical random variable represents outcomes from a category or label.

26
Q

Example: What is the probability of selecting a red candy from a bag containing red, green, and blue candies?

A

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.

27
Q

Types of discrete probability distributions

A

Bernoulli Distribution, Binomial Distribution, Poisson Distribution, Geometric Distribution, Hypergeometric Distribution, Negative Binomial Distribution, Discrete Uniform Distribution, Categorical Distribution, Multinomial Distribution.

28
Q

What is a Bernoulli Distribution?

A

A Bernoulli distribution represents a single trial with two possible outcomes – success (usually denoted as 1) or failure (usually denoted as 0).

29
Q

Example: What is the probability of flipping a coin and getting heads?

A

The probability of flipping a coin and getting heads is 0.5 in a fair coin toss.

30
Q

What is a Binomial Distribution?

A

The binomial distribution represents the number of successes in a fixed number of independent Bernoulli trials.

31
Q

Example: If you roll a fair six-sided die five times, what’s the probability of getting exactly two fives?

A

The probability of getting exactly two fives in five rolls of a fair six-sided die can be calculated using the binomial distribution formula.

32
Q

What is a Poisson Distribution?

A

The Poisson distribution models the number of events occurring in a fixed interval of time or space when events happen randomly and independently.

33
Q

Example: In a busy restaurant, what is the probability of receiving exactly three customer complaints in an hour?

A

The probability of receiving exactly three customer complaints in an hour can be calculated using the Poisson distribution formula.

34
Q

What is a Geometric Distribution?

A

The geometric distribution represents the number of trials needed for the first success in a sequence of independent Bernoulli trials.

35
Q

Example: How many times do you need to roll a die to get the first six?

A

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.

36
Q

What is a Hypergeometric Distribution?

A

The hypergeometric distribution represents the number of successes in a sample drawn without replacement from a finite population.

37
Q

Example: In a deck of cards, if you draw five cards without replacement, what’s the probability of getting exactly two Aces?

A

The probability of getting exactly two Aces when drawing five cards from a deck without replacement can be calculated using the hypergeometric distribution formula.

38
Q

What is a Negative Binomial Distribution?

A

The negative binomial distribution represents the number of trials needed to achieve a fixed number of successes in a sequence of independent Bernoulli trials.

39
Q

Example: How many times do you need to flip a coin to get three heads?

A

The expected number of coin flips needed to get three heads in a row can be calculated using the negative binomial distribution formula.

40
Q

What is a Discrete Uniform Distribution?

A

The discrete uniform distribution represents outcomes that are equally likely among a finite set of values.

41
Q

Example: If you roll a fair six-sided die, what’s the probability of getting a three?

A

The probability of rolling a three on a fair six-sided die is 1/6.

42
Q

What is a Categorical Distribution?

A

The categorical distribution represents outcomes from a category or label.

43
Q

Example: In a survey, what is the probability of someone choosing “Yes” from the options “Yes,” “No,” and “Maybe”?

A

The probability of someone choosing “Yes” from the options can be determined based on the categorical distribution probabilities.

44
Q

What is a Multinomial Distribution?

A

The multinomial distribution represents the probabilities of outcomes in a categorical experiment with more than two categories.

45
Q

Example: In a bag of colored marbles, what’s the probability of drawing two red, three blue, and one green marble in six draws?

A

The probability of drawing the specified combination of marbles can be calculated using the multinomial distribution formula.