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What is the definition of a random variable?
A numerical outcome from a random event.
What is the definition of expected value?
The average outcome over many trials; the long-run average.
What is the definition of variance?
A measure of how spread out the values of a random variable are around the mean.
What is the definition of standard deviation?
The square root of the variance; measures spread in the same units as the random variable.
Define a discrete random variable.
A random variable with countable values.
What is the formula for the expected value (mean) of a discrete random variable?
E(X) = \sum x \cdot P(X=x)
What is the formula for the variance of a discrete random variable?
Var(X) = \sum (x - E(X))^2 \cdot P(X=x)
What is the formula for the standard deviation of a discrete random variable?
SD(X) = \sqrt{Var(X)}
Given a probability distribution, how do you calculate E(X)?
Multiply each value of X by its probability and sum the results.
How is standard deviation related to variance?
Standard deviation is the square root of the variance.
Explain the concept of expected value.
It's the long-run average outcome if you repeat an experiment many times. It doesn't have to be a possible value of X.
Explain the concept of variance.
Variance quantifies the spread of data points around the mean. A higher variance indicates greater variability.
Explain the concept of standard deviation.
Standard deviation measures the typical deviation of values from the mean. It's in the same units as the original data, making it easier to interpret than variance.
Explain how probabilities are used in calculating the mean of a discrete random variable.
Probabilities act as weights, influencing how much each outcome contributes to the overall average. Outcomes with higher probabilities have a greater impact on the mean.
Explain why we square the differences from the mean when calculating variance.
Squaring makes all deviations positive (so positive and negative deviations don't cancel out) and emphasizes larger deviations.