All Flashcards
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 are the differences between variance and standard deviation?
Variance: Average squared distance from the mean, units are squared. | Standard Deviation: Square root of variance, units are the same as the data.
What are the differences between a parameter and a random variable?
Parameter: Describes a population. | Random Variable: Assigns numerical values to outcomes of random phenomena.
What are the differences between discrete and continuous random variables?
Discrete: Countable values (e.g., number of texts). | Continuous: Any value within a range (e.g., height).
What are the differences between calculating variance and standard deviation?
Variance: Requires summing the squared differences from the mean, weighted by probabilities. | Standard Deviation: Simply taking the square root of the calculated variance.
What are the differences between interpreting variance and standard deviation?
Variance: Indicates the average squared deviation from the mean; harder to directly interpret. | Standard Deviation: Indicates the typical deviation from the mean in original units; easier to interpret in context.
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.