Discrete Probability Distributions Practice Flashcards

Flashcards covering the definitions, rules, and mathematical formulas for discrete probability distributions, including mean, variance, and standard deviation calculations.

Discrete Probability Distribution

A listing of all possible outcomes of an experiment for a discrete random variable along with the relative frequency, which represents the probability.

Mutually Exclusive

A rule for discrete probability distributions stating that each outcome in the distribution is independent of and cannot occur at the same time as all other outcomes.

Probability Range Rule

A general rule for probability distributions stating that the probability of each outcome, denoted as $P(x)$, must be between $0$ and $1$.

Sum of Probabilities Rule

A fundamental requirement for discrete probability distributions where the sum of the probabilities for all outcomes must be equal to $1$.

Mean ($\mu$)

A weighted average of the outcomes of the random variables that comprise a distribution, also known as the expected value ($E(x)$), calculated as the summation of each $x$ variable multiplied by its relative frequency or probability: $\mu = \sum x \cdot P(x)$.

Expected Value ($E(x)$)

Another term for the mean ($\mu$) of a discrete probability distribution.

Variance ($\sigma^2$)

A measure of dispersion calculated by taking each $x$ value, subtracting the mean ($\mu$), squaring that difference, multiplying it by the relative frequency ($P(x)$), and then summing those values: $\sum (x - \mu)^2 \times P(x)$.

Standard Deviation ($\sigma$)

A measure of variation obtained by taking the square root of the variance.

Relative Frequency

The frequency of a particular bin or outcome divided by the total frequency.

Atlanta Call Center Stats

A call center with a mean number of rings of $3.15$, a variance of $1.5275$, and a standard deviation of approximately $1.24$.

Boston Call Center Stats

A call center showing a mean number of rings of $1.9$, a variance of $0.69$, and a standard deviation of approximately $0.8307$.

Managerial Performance Conclusion

Based on the comparison, the Boston call center is deemed more effective because it has a lower average time to answer calls (mean) and lower variability (standard deviation), indicating more consistency.