1. Binomial Distribution

Definition:

The binomial distribution applies to independent repeated trials where each trial has only two possible outcomes (success or failure). Let the probability of success in each trial be ( p ), the probability of failure be ( 1 - p ), and the number of trials be ( n ).

Standard Deviation Formula:

$$
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
$$

Explanation:

• ( n ): The number of trials.
• ( p ): The probability of success.
• ( 1 - p ): The probability of failure.

Example:

Suppose you toss a fair coin ( n ) times and calculate the standard deviation for the number of heads (success).

• ( n = 10 )
• ( p = 0.5 )
• ( 1 - p = 0.5 )

Calculation:

$$
\sigma = \sqrt{10 \cdot 0.5 \cdot 0.5} = \sqrt{2.5} \approx 1.58
$$

2. Hypergeometric Distribution

Definition:

The hypergeometric distribution describes sampling without replacement from a finite population. The population size is ( N ), where ( K ) items are considered successes (e.g., red balls), and the remaining ( N - K ) are failures (e.g., blue balls). A sample of ( n ) is drawn, and we focus on the number of successes in the sample.

Standard Deviation Formula:

$$
\sigma = \sqrt{n \cdot \frac{K}{N} \cdot \frac{N - K}{N} \cdot \frac{N - n}{N - 1}}
$$

Explanation:

• ( N ): The population size.
• ( K ): The number of successes.
• ( n ): The sample size.

Example:

Suppose there are ( N = 20 ) balls, of which ( K = 5 ) are red, and the remaining are blue. You draw ( n = 4 ) balls. The standard deviation of the number of red balls in the sample is:

Calculation:

$$
\sigma = \sqrt{4 \cdot \frac{5}{20} \cdot \frac{15}{20} \cdot \frac{16}{19}} \approx 0.87
$$

3. Normal Distribution

Definition:

The normal distribution is a key continuous probability distribution, and many natural phenomena approximately follow it. The standard deviation measures the spread of data around the mean.

Standard Deviation Formula for Sample Data:

$$ \sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}}\ $$

Explanation:

• $x_i$: The i-th data point.
• $\mu$: The mean of the data.
• $n$: The number of data points.

Example:

Suppose you have a set of measurements: $$ x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4, x_5 = 5 $$

Mean Calculation:

$$
\mu = \frac{1 + 2 + 3 + 4 + 5}{5} = 3
$$

Standard Deviation Calculation: $$
\sigma = \sqrt{\frac{(1-3)^2 + (2-3)^2 + (3-3)^2 + (4-3)^2 + (5-3)^2}{5}} = \sqrt{2} \approx 1.41
$$

4. Poisson Distribution

Definition:

The Poisson distribution is used to describe the number of events occurring within a fixed interval of time or space, with the parameter $\lambda$, which represents the average number of occurrences.

Standard Deviation Formula:

$$
\sigma = \sqrt{\lambda}
$$

Explanation:

• $\lambda$ : The average number of occurrences.

Example:

If a store receives an average of $\lambda$ = 5 customers per hour, the standard deviation for the number of customers is:

$$
\sigma = \sqrt{5} \approx 2.24
$$

5. Geometric Distribution

Definition:

The geometric distribution describes the number of failures before the first success, with the probability of success ( p ).

Standard Deviation Formula:

$$
\sigma = \frac{\sqrt{1 - p}}{p}
$$

Explanation:

• ( p ): The probability of success.

Example:

If the probability of making a successful basketball shot is ( p = 0.3 ), the standard deviation for the number of attempts before the first success is:

$$
\sigma = \frac{\sqrt{1 - 0.3}}{0.3} \approx 2.19
$$

6. Exponential Distribution

Definition:

The exponential distribution describes the time between events, with the parameter $\lambda$, representing the rate of occurrence.

Standard Deviation Formula:

$$
\sigma = \frac{1}{\lambda}
$$

Explanation:

• $\lambda$: The rate of occurrence.

Example:

If a device fails on average $\lambda$ = 2 times per hour, the standard deviation of the time between failures is:

$$
\sigma = \frac{1}{2} = 0.5 \text{ hours}
$$