Chapter 5: Discrete Probability Distributions
Probability Distributions
Terms
Variable: Characteristic that can assume different values
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Discrete Variables: Values are discrete quantitative units
Example: Number of wheels
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Continuous Variables: Values are a point within a continuous range
Example: Speed
Random Variable: Variable whose value is determined randomly
Discrete Probability Distribution
Discrete probability distribution:
- Lists the discrete values a random variable can assume, and their corresponding probability
- Specific outcomes are notated as $X$; the probability of $X$ occurring is notated as $P(X)$
- Can be determined theoretically or empirically
- Theoretically:
- Where there are $n$ potential discrete values, and each has an equally random chance, the probability for each discrete value is $\frac {1} {n}$
- Example: Discrete Probability Distribution for rolling a standard die
$X$ 1 2 3 4 5 6 $P(X)$ $\frac {1} {6}$ $\frac {1} {6}$ $\frac {1} {6}$ $\frac {1} {6}$ $\frac {1} {6}$ $\frac {1} {6}$ - For a compound event, which includes any of $r$ number of potential values, the probability for the compound event is $\frac {r} {n}$
- Example: Discrete Probability Distribution for rolling an odd or even
$X$ Odd Even $P(X)$ $\frac {1} {2}$ $\frac {1} {2}$
- Where there are $n$ potential discrete values, and each has an equally random chance, the probability for each discrete value is $\frac {1} {n}$
- Empirically:
- Determined by observing the outcome of numerous trials. For $n$ trials, the probability for each value that occurred $r$ times is $\frac {r} {n}$
- Theoretically:
Discrete Probability Distribution Requirements
The sum of all probabilities within a sample space must be 1.
\[\Sigma P(X)=1\]The probability of each event cannot be lower than 0 or greater than 1.
\[0 \le P(X) \le 1\]Mean, Variance, Standard Deviation, Expectation
Mean
Mean:
To calculate the mean for discrete quantitative data, sum the products of each value with its frequency.
\[\mu = \Sigma X \cdot P(X)\]Variance, Standard Deviation
Variance:
To calculate the variance for discrete quantitative data, sum the products of the square of the difference between each value and the mean with the value’s frequency.
\[\sigma ^2 = \Sigma \big( ( X - \mu ) ^2 \cdot P(X) \big)\]Variance shortcut formula:
There is a shortcut formula for calculating the variance:
\[\sigma ^2 = \Sigma \big( X ^2 \cdot P(X) \big) - \mu ^2\]Standard deviation:
The standard deviation is the square root of the variance; $\sqrt{\sigma^2}$, or:
\[\sqrt{\Sigma \big( X ^2 \cdot P(X) \big) - \mu ^2}\]Expectation
Expectation: The expected return from a trial
- Similar formula as mean, however, instead of using the raw output value, it adjusts the output for an expense incurred at input
- Example: If a lottery ticket costs 1 and has a chance of winning 1000, the mean would calculate the potential results as 0 and 1000; the expectation would adjust for the input cost, and calculate the potential results as -1 and 999.
Binomial Distribution
Binomial Distribution
Binomial Distribution:
- Takes a trial repeated identically $n$ times; each trial has two potential outcomes - success or failure; the rate of success for an individual trial is known; calculates the probability that over $n$ trials there will be $X$ successes.
Formula
Formula for calculating a binomial distribution:
where
- $n$ is the number of trials;
- $p$ is the probability of success;
- $q$ is the probability of failure ($q=1-p$); and
- $X$ is the amount of successes;
the probability that in $n$ trials specifically $X$ trials will be successful is calculated as follows:
\[P(X) = \space _n C _x \cdot p ^X \cdot q ^{n-X}\]Binomial Distribution Table
Binomial Distribution Table:
- There is a precalculated table which lists the probabilities of binomial distributions.
- The table takes $n$, $X$, and $P(S)$, and lists $P(X)$
Mean, Variance, Standard Deviation for Binomial Distribution
Binomial Distributions have specific shortcut formulas for calculating their statistical measures:
- Mean
- Variance
- Standard Deviation