Summary Measures of Probability Distributions and the Normal Distribution

Vocabulary flashcards covering summary measures of probability distributions, Excel computational procedures, properties of the normal distribution, and continuous density functions.

Mean of a Probability Distribution ($\mu$ or $E(X)$)

A weighted sum of the possible values, weighted by their probabilities: $\mu = E(X) = \sum_{i=1}^n x_i p(x_i)$. It is also called the expected value of $X$.

Variance of a Probability Distribution ($\sigma^2$ or $\text{Var}(X)$)

A weighted sum of the squared deviations of the possible values from the mean, weighted by probabilities: $\sigma^2 = \text{Var}(X) = \sum [x_i - E(X)]^2 p(x_i)$.

Computing Formula for Variance

An algebraic computing formula for the variance of a probability distribution given by $\sigma^2 = \sum x_i^2 p(x_i) - \mu^2$.

Standard Deviation ($\sigma$ or $\text{Stdev}(X)$)

A measure of variability calculated as the square root of the variance: $\sigma = \text{Stdev}(X) = \sqrt{\text{Var}(X)}$.

=SUMPRODUCT(Market_return,Probability)

The Excel formula entered in cell B11 to calculate the mean return in Example 5.2.

=(C4-Mean)^2

The Excel formula entered in cell D4 (and copied through D8) to calculate squared deviations from the mean for computing variance in Example 5.2.

=SUMPRODUCT(Market_return,Market_return,Probability)-Mean^2

The simplified Excel formula used in cell B12 to directly calculate variance without needing individual squared deviation columns.

=SQRT(Variance)

The Excel formula entered in cell B13 to calculate the standard deviation of market return in Example 5.2.

Normal Distribution

A continuous distribution that forms the basis of the symmetric bell-shaped curve and is the single most important distribution in statistics.

Two-Parameter Family

A classification describing the normal distribution, where any particular distribution is fully specified by two parameters: the mean and standard deviation.

Shifting and Spreading of the Normal Curve

The geometric effect of the parameters on the normal distribution: changing the mean shifts the curve to the right or left, while changing the standard deviation makes the curve more or less spread out.

Continuous Distribution

A probability distribution that features a continuum of possible values (such as all values between $0$ and $100$, or all values greater than $0$), over which the total probability of $1$ is spread rather than assigned to individual values.

Density Function

The function that spreads the total probability of $1$ over a continuum for continuous distributions, acting like a histogram where higher values indicate more likely regions.