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--- b:head_first_statistics:geometric_binomial_and_poisson_distributions [2023/10/19 17:45] – [Geometric Binomial and Poisson Distributions] hkimscil
+++ b:head_first_statistics:geometric_binomial_and_poisson_distributions [2023/10/19 19:00] (current) – [Geometric Binomial and Poisson Distributions] hkimscil
@@ Line 6: / Line 6: @@
 E\left[ X \right] & = \frac{1}{p} \\
 V\left[ X \right] & = \frac{q}{p^2} \\
-\end{align*}
+\\
+\text{Binomial Distribution:  } \;\;\; \text{X} & \thicksim B(n, p) \\
-\begin{align*}
-\text{Binomial Distribution:  } \;\;\; \text{X} & \thicksim B(p) \\
 p(X = r) & = \binom{n}{r} \cdot p^{r} \cdot q^{n-r} \\
 E\left[ X \right] & = {n}{p} \\
 V\left[ X \right] & = {n}{p}{q} \\
-\end{align*}
+\\
-\begin{align*}
 \text{Poisson Distribution:  } \;\;\; \text{X} & \thicksim P( \lambda ) \\
-p(X = r) & = \binom{n}{r} \cdot p^{r} \cdot q^{n-r} \\
+P(X=r) & = e^{- \lambda} \cdot \dfrac{\lambda^{r}} {r!} \\
-E\left[ X \right] & = {n}{p} \\
+E\left[ X \right] & = \lambda \\
-V\left[ X \right] & = {n}{p}{q} \\
+V\left[ X \right] & = \lambda \\
 \end{align*}