Discrete Probability Distribution Visualization
Observe probability density distribution changes in real-time through interactive parameters
1. Definition of Discrete Probability Distribution
Suppose the possible values of random variable $$X$$ are $$\{x_1, x_2, \dots, x_k, \dots\}$$, with corresponding probabilities $$\{p_1, p_2, \dots, p_k, \dots\}$$.
That is:
$$p_k \equiv P(X = x_k)$$
Then $$X$$ is called a discrete random variable, and its probability distribution can be expressed as:
$$
\begin{array}{c|ccccc}
X & x_1 & x_2 & \dots & x_k & \dots \\
\hline
p & p_1 & p_2 & \dots & p_k & \dots
\end{array}
$$
Which must satisfy:
- $$p_k \ge 0$$
- $$\sum_{k} p_k = 1$$
Current Example: Binomial Distribution
Describes the probability of $$k$$ successes in $$n$$ independent trials, where $$p$$ is the probability of success in a single trial.
$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$