2.4.1 Discrete random variables

• A discrete random variable is one that can only take a finite or countably infinite number of distinct values.
• Discrete random variables can take any real number within a given range.

  False
• What function is used to calculate probabilities for discrete random variables?

  Probability Mass Function
• The PMF for a discrete random variable describes the likelihood of each possible value occurring.
• The CDF for a discrete random variable gives the cumulative probability up to a specific value.
• How is the expected value of a discrete random variable calculated?

  Sum of values multiplied by probabilities
• The standard deviation of a discrete random variable is the square root of its variance.
• Bernoulli and Binomial distributions are examples of discrete random variables.
• What is the probability mass function of a Bernoulli distribution?

  P(X = x) = p^x(1 - p)^{1 - x}</latex>
• The Binomial distribution models the number of successes in a fixed number of trials.
• What does a probability distribution describe for a discrete random variable?

  Likelihood of each possible value
• Order the steps to calculate probabilities for a discrete random variable:

  1️⃣ Identify the possible values of the variable

  2️⃣ Determine the Probability Mass Function (PMF)

  3️⃣ Calculate the probability for each value using the PMF

  4️⃣ Verify that the probabilities sum to 1
• For a discrete random variable X</latex>, the probability $P(X = x)$ represents the likelihood that $X$ takes the value x.
• Continuous random variables can take any value within a given range.
• The probability distribution for a continuous random variable is described by a Probability Density Function (PDF).
• Match the random variable type with its example:

  Discrete ↔️ Number of heads in coin flips

  Continuous ↔️ Height of students
• What is the key difference between discrete and continuous random variables in terms of values?

  Finite vs infinite within a range
• The probability at a single point for a continuous random variable is always zero.
• Order the steps to create a probability distribution table for a discrete random variable:

  1️⃣ List all possible values of the variable

  2️⃣ Calculate the probability for each value

  3️⃣ Create a table with values and probabilities

  4️⃣ Verify that the probabilities sum to 1
• What is the shape of the probability mass function for a Bernoulli distribution?

  Two possible values with probabilities
• The probability distribution for a discrete random variable is called the Probability Mass Function (PMF).
• The probability at a single point for a continuous random variable is zero.
• What type of random variable is the number of cars passing a point in an hour?

  Discrete
• What is a discrete random variable?

  Finite or countably infinite values
• A discrete random variable can only take a finite or countably infinite number of distinct values
• What type of values can a continuous random variable take?

  Any real number within a range
• A continuous random variable can take any value within a given range
• Match the feature with its description for discrete and continuous random variables:

  Values ↔️ Finite or countably infinite ||| Infinite within a range

  Measurability ↔️ Countable ||| Measured

  Probability Distribution ↔️ PMF ||| PDF

  Probability at a Point ↔️ Non-zero ||| Zero
• The probability at a single point for a continuous random variable is always zero.
• Give an example of a discrete random variable.

  Number of heads in coin flips
• The number of cars passing a point in an hour is an example of a discrete random variable
• What is an example of a continuous random variable?

  Time to complete a race
• The Probability Mass Function (PMF) is used to calculate probabilities for discrete random variables.
• The Probability Mass Function (PMF) is denoted by $P(X = x)$ and gives the probability of each possible value
• What is $P(X = 1)$ in the example of flipping a coin twice?


  0.5
• In the coin flip example, the probability of getting exactly one head is 50%.
• Steps to calculate probabilities for a discrete random variable using the PMF

  1️⃣ Identify the possible values of $X$

  2️⃣ Determine the probability of each value $P(X = x)$

  3️⃣ Verify that the probabilities sum to 1: $\sum P(X = x) =$$1$
• What is a Probability Mass Function (PMF) used for?

  Describing the probability distribution of a discrete random variable
• The PMF assigns a probability to each possible value of a discrete random variable.
• The sum of probabilities in a PMF must equal 1.