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Answer: Bayes Theorem can only be applied to discrete random variables, such that continuous random variables must be transformed into their discrete equivalents
## Explanation Let's analyze each option: **Option A: TRUE** - The cumulative distribution function (CDF) F(x) must satisfy F(−∞) = 0 and F(+∞) = 1.0 for both discrete and continuous random variables. This is a fundamental property of CDFs. **Option B: TRUE** - CDFs are non-decreasing functions for both discrete and continuous random variables. As x increases, F(x) cannot decrease. **Option C: TRUE** - For continuous random variables, the probability of any specific value is zero (Pr[X = a] = 0). We can only talk about probabilities over intervals. **Option D: FALSE** - This is the incorrect statement. Bayes Theorem can be applied to both discrete and continuous random variables. For continuous variables, we use probability density functions (PDFs) instead of probability mass functions (PMFs). There's no requirement to transform continuous variables into discrete equivalents to apply Bayes Theorem. Bayes Theorem for continuous variables is expressed using probability density functions: $$f(\theta|X) = \frac{f(X|\theta)f(\theta)}{\int f(X|\theta)f(\theta)d\theta}$$ Where f(θ|X) is the posterior density, f(X|θ) is the likelihood function, and f(θ) is the prior density.
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Let f(x) represent a probability function (which is called a probability mass function, p.m.f., for discrete random variables and a probability density function, p.d.f., for continuous variables) and let F(x) represent the corresponding cumulative distribution function (CDF); in the case of the continuous variable, F(x) is the integral (aka, anti-derivative) of the pdf. Each of the following is true about these probability functions EXCEPT which is false?
A
The limits of a cumulative distribution function (CDF) must be zero and one; i.e., F(−∞) = 0 and F(+∞) = 1.0
B
For both discrete and continuous random variables, the cumulative distribution function (CDF) is necessarily a non-decreasing function.
C
In the case of a continuous random variable, we cannot talk about the probability of a specific value occurring; e.g., Pr[R = +3.00%] is meaningless
D
Bayes Theorem can only be applied to discrete random variables, such that continuous random variables must be transformed into their discrete equivalents