What is the cardinality of the following set? p(R\6)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Discrete math
![Title: Cardinality of Sets
**Question:**
What is the cardinality of the following set?
\[ \mathscr{P} \left( \mathbb{R} \setminus \overline{\mathbb{Q}} \right) \]
- ⭕ ℵ₀
- ⭕ ℵ₁
- ⭕ ℵ₂
- ⭕ ℵ₃
- ⭕ None of the above.
**Explanation:**
The problem asks for the cardinality of the power set of the set of real numbers excluding the rational numbers. The power set of a given set has a cardinality of \(2^{\text{cardinality of the set}}\). The cardinality of the real numbers without the rational numbers (irrational numbers) remains the same as that of the real numbers, which is \( \mathfrak{c} \) (the cardinality of the continuum). The power set, therefore, has a cardinality of \(2^{\mathfrak{c}}\), which is larger than ℵ₁, ℵ₂, or any countable infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fafcae178-18ab-4874-919f-bfc7f64031d3%2Fa9e8d45e-865c-4bce-9245-ea8fa7a07750%2F2bakiph_processed.png&w=3840&q=75)
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