According to the Addition Theorem of Probability:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

By definition, two events

\(A\) and \(B\) are \lt strong\gt mutually exclusive\lt /strong\gt (disjoint) if they cannot occur simultaneously, which means their intersection is impossible: \[ A \cap B = \emptyset \implies P(A \cap B) = 0 \]

Substituting

\(P(A \cap B) = 0\) into the addition theorem yields: \[ P(A \cup B) = P(A) + P(B) \]

Detailed Analysis of Probability Distributions Properties:

• Normal Distribution: Mean = $\mu$, Variance = $\sigma^2$ (independent parameters; generally $\mu \ne \sigma^2$).
• Binomial Distribution: Mean = $np$, Variance = $npq$ (since $q \lt 1$, Mean > Variance).
• Poisson Distribution: Defined by single parameter $\lambda$. Mean = $\lambda$ and Variance = $\lambda$. Hence, Mean = Variance holds strictly for Poisson distribution only.
• F-distribution: Mean = $d_2 / (d_2 - 2)$ for $d_2 \gt 2$, Variance is a complex function of degrees of freedom.
• Chi-square Distribution: Mean = $k$, Variance = $2k$ (Variance = $2 \times$ Mean).

The mathematical expectation $E(X)$ of a random variable $X$ represents the weighted average or mean (first raw moment) of its probability distribution. Option analysis: (A) Mean is the exact statistical definition of expectation. (B) Median is the middle value where cumulative probability is $0.5$. (C) Variance is $E[(X - \mu)^2]$, the second central moment. (D) Standard deviation is the square root of variance.

Binomial (B) and Poisson (C) distributions are discrete probability distributions based on discrete count data. Normal (A), F-distribution (D), and Chi-square (E) distributions are continuous probability distributions defined on continuous random variables. Option analysis: (A) Includes discrete distributions (B) and (C). (B) Includes discrete distribution (B). (C) Includes discrete distributions (B) and (C). (D) (A), (D) and (E) correctly groups all continuous distributions.