4.3 Mean or Expected Value and Standard Deviation
Mean or Expected Value: \(\mu=\sum_{x \in X} x P(x)\)
Standard Deviation: \(\sigma=\sqrt{\sum_{x \in X}(x-\mu)^2 P(x)}\)
4.4 Binomial Distribution
\(X \sim B(n, p)\) 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 \(x=0,1,2,3, \ldots, n\)
\(p=\) 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 \(\mu=n p\). The standard deviation of \(X\) is \(\sigma=\sqrt{n p q}\).
4.5 Geometric Distribution
\(X \sim \mathrm{G}(p)\) 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 \(x=1,2,3, \ldots\)
\(p=\) 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 is \(\mu=\frac{1}{p}\).
The standard deviation is \(\sigma=\sqrt{\frac{1-p}{p^2}}=\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}\).
4.6 Hypergeometric Distribution
\(X \sim H(r, b, n)\) 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 \(x= 0,1, \ldots\), up to the size of the group of interest. (The minimum value for \(X\) may be larger than zero in some instances.)
\(n \leq r+b\)
The mean of \(X\) is given by the formula \(\mu=\frac{n r}{r+b}\) and the standard deviation is \(=\sqrt{\frac{r b n(r+b-n)}{(r+b)^2(r+b-1)}}\).
4.7 Poisson Distribution
\(X \sim P(\mu)\) means that \(X\) has a Poisson probability distribution where \(X=\) the number of occurrences in the interval of interest.
\(X\) takes on the values \(x=0,1,2,3, \ldots\)
The mean \(\mu\) is typically given.
The variance is \(\sigma^2=\mu\), and the standard deviation is \(\sigma=\sqrt{\mu}\).
The probability of having exactly \(x\) successes in \(r\) trials is \(P(X=x)=\left(e^{-\mu}\right) \frac{\mu^x}{x!}\).
When \(P(\mu)\) is used to approximate a binomial distribution, \(\mu=n p\) where \(n\) represents the number of independent trials and \(p\) represents the probability of success in a single trial.