4.7 Introduction to Random Variables and Probability Distributions

• What is a random variable?

  Outcome of a random phenomenon
• Match the type of random variable with its example:

  Discrete ↔️ Number of coin flips

  Continuous ↔️ Height of a person
• A discrete random variable can take on a finite or countably infinite number of values
• A discrete random variable has a finite or countably infinite number of values
• What does a probability distribution describe?

  Likelihood of random variable values
• What does a probability mass function (PMF) assign probabilities to?

  Possible values of a discrete variable
• What is a random variable?

  Numerical outcome of random phenomenon
• What is the key difference between discrete and continuous random variables?

  Type of values they take
• What does a discrete probability distribution list?

  Probabilities for each possible value
• What is the purpose of a probability mass function (PMF)?

  Assign probabilities to discrete values
• Match the property with the uniform distribution $U(0, 1)$:


  Density Function ↔️ $f(x) =$$1$

  Range ↔️ $0 \leq x \leq 1$

  Total Area Under Curve ↔️ 1
• A CDF is applicable for both discrete and continuous random variables
• A continuous random variable can take on any value within a given range


  True
• What type of values can a continuous random variable take?

  Real numbers within a range
• For continuous random variables, the total area under the probability density function equals 1


  True
• Steps to create a PMF for the number of heads in two coin flips:

  1️⃣ Identify possible values (0, 1, 2)

  2️⃣ Calculate the probability of each value

  3️⃣ Summarize probabilities in a table
• A discrete random variable can take a finite or countable number of values
• A probability distribution describes the likelihood of different values
• A continuous probability distribution uses a probability density function to represent probabilities over a range
• In a PMF, all probabilities must be between 0 and 1 and sum up to 1.


  True
• What does a cumulative distribution function (CDF) give?

  Probability of $X \leq x$
• Match the probability distribution with its property:

  Binomial ↔️ Fixed number of independent trials

  Uniform ↔️ Equally likely outcomes within a range

  Normal ↔️ Bell-shaped curve defined by mean and standard deviation
• Steps to distinguish between discrete and continuous random variables:

  1️⃣ Determine the number of possible values

  2️⃣ Identify the type of values (whole or real numbers)

  3️⃣ Check if values are countable
• Match the type of probability distribution with its description:

  Discrete ↔️ Lists probabilities for each value

  Continuous ↔️ Uses a probability density function
• What must the probabilities in a PMF sum up to?

  1
• A continuous random variable can take any value within a given range.


  True
• In a discrete probability distribution, the sum of all probabilities equals 1.

  True
• The total area under a probability density function (PDF) equals 1.


  True
• A probability density function (PDF) defines the likelihood of a continuous random variable falling within a given range
• A CDF is always non-decreasing and ranges from 0 to 1.


  True