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Discrete Random Variable Distribution Experiment

Mathematical visualization of Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)

Theoretical Definitions

Discrete Random Variable

Let $$X$$ be a discrete random variable defined on the sample space $$\Omega$$, with possible values $$x_1, x_2, \ldots, x_k, \ldots$$.

1. Probability Mass Function (PMF)

PMF is defined as the probability that a random variable takes a specific value:

$$f_X(k) = P(X = k)$$

Properties: $$f_X(k) \ge 0$$ and $$\sum_{k} f_X(k) = 1$$

2. Cumulative Distribution Function (CDF)

CDF is defined as the probability that a random variable is less than or equal to a certain value:

$$F_X(x) = P(X \le x) = \sum_{k \le x} P(X = k)$$

For discrete variables, CDF is a right-continuous step function satisfying $$0 \le F_X(x) \le 1$$

CDF and Integration Relationship

Although discrete distributions do not use traditional integration, CDF can be viewed as the cumulative sum of PMF:

$$F_X(x) = \sum_{k=-\infty}^{x} f_X(k)$$

For continuous distributions, CDF is the integral of PDF: $$F_X(x) = \int_{-\infty}^{x} f_X(t) dt$$

Binomial Distribution Model

In $$n$$ independent Bernoulli trials, the number of successes $$X$$ follows a binomial distribution $$X \sim \text{Binomial}(n, p)$$:

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$
$$n$$: Number of trials
$$p$$: Probability of success
$$k$$: Number of successes
$$0 \le k \le n$$

Expected Value: $$E[X] = np$$

Variance: $$\text{Var}(X) = np(1-p)$$

Probability Mass Function (PMF)

Probability that the discrete random variable takes each possible value

$$f(k) = P(X = k)$$
X-axis: $$k$$ (Number of successes)
Y-axis: $$P(X=k)$$ (Probability)

Cumulative Distribution Function (CDF)

Cumulative probability that the random variable is less than or equal to a certain value

$$F(x) = P(X \le x) = \sum_{k \le x} P(X=k)$$
X-axis: $$x$$
Y-axis: $$F(x)$$ (Cumulative probability)

Experiment Instructions

Parameter Adjustment

Adjust the number of trials $$n$$ and success probability $$p$$ to observe distribution changes.

Chart Interaction

Hover over to view exact values and observe the relationship between PMF and CDF.

Mathematical Understanding

The height of each step in CDF equals the PMF value at that point, understanding the cumulative nature of discrete distributions.