4.2 Mean or Expected Value and Standard Deviation

Mean or Expected Value: $\mu =\underset{x\in X}{{\displaystyle \Sigma }}xP(x)$

Standard Deviation: $\sigma =\sqrt{\underset{x\in X}{{{\displaystyle \Sigma }}^{\text{}}}{(x-\mu )}^{2}P(x)}$

4.3 Binomial Distribution (Optional)

X ~ B(n, p) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p.

X = the number of successes in n independent trials

n = the number of independent trials

X takes on the values x = 0, 1, 2, 3, . . . , n

p = the probability of a success for any trial

q = the probability of a failure for any trial

p + q = 1; q = 1 – p

The mean of X is μ = np. The standard deviation of X is σ = $\sqrt{npq}$.

4.4 Geometric Distribution (Optional)

X ~ G(p) means that the discrete random variable X has a geometric probability distribution with probability of success in a single trial p.

X = the number of independent trials until the first success

X takes on the values x = 1, 2, 3, . . .

p = the probability of a success for any trial

q = the probability of a failure for any trial p + q = 1

The mean is μ = $\frac{1}{p}$.

The standard deviation is σ = $\sqrt{\frac{1–p}{p^2}}$ = $\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}$ .

4.5 Hypergeometric Distribution (Optional)

X ~ H(r, b, n) means that the discrete random variable X has a hypergeometric probability distribution with r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample.

X = the number of items from the group of interest that are in the chosen sample, and X may take on the values x = 0, 1, . . . , up to the size of the group of interest. The minimum value for X may be larger than zero in some instances.

n ≤ r + b

The mean of X is given by the formula μ = $\frac{nr}{r\text{ + }b}$ and the standard deviation is = $\sqrt{\frac{rbn(r\text{ + }b-n)}{{(r\text{ + }b)}^2(r\text{ + }b-\text{1)}}}$.

4.6 Poisson Distribution (Optional)

X ~ P(μ) means that X has a Poisson probability distribution where X = the number of occurrences in the interval of interest.

X takes on the values x = 0, 1, 2, 3, . . .

The mean μ is typically given.

The variance is σ² = μ, and the standard deviation is

When P(μ) is used to approximate a binomial distribution, μ = np where n represents the number of independent trials and p represents the probability of success in a single trial.