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Question: Give a characteristic of a uniform random variable. Choose the correct answer. A. A uniform random variable is more likely to assume a value in the specified interval than it is to assume a value in any other interval of equal size, OB. A uniform random variable has a probability distribution that can be approximated by a normal distribution O c. A uniform
| **Give a characteristic of a uniform random variable.** | |
| --- | --- |
| Choose the correct answer. | |
| A. A uniform random variable is more likely to assume a value in the specified interval than it is to assume a value in any other interval of equal size. | |
| B. A uniform random variable has a probability distribution that can be approximated by a normal distribution. | |
| C. A uniform random variable has a probability distribution that can be approximated by an exponential probability distribution. | |
| D. A uniform random variable is just as likely to assume a value in one interval as it is to assume a value in any other interval of equal size. | |
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Answer
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Step 1:Let me solve this step by step:
Step 2:: Understand the Uniform Random Variable
A uniform random variable is a continuous probability distribution where every value within a specified interval has an equal probability of occurring. This means the probability density is constant across the entire interval.
Step 3:: Analyze the Given Options
Let's carefully examine each option: - Option A suggests unequal probabilities (incorrect) - Option B suggests approximation by normal distribution (incorrect) - Option C suggests approximation by exponential distribution (incorrect) - Option D suggests equal probabilities across intervals
Step 4:: Confirm the Characteristic
The key characteristic of a uniform random variable is that ALL intervals of equal width within its range have EQUAL probability of occurrence.
Step 5:: Mathematical Representation
For a uniform distribution $$X \sim U(a,b)$$, the probability density function is:
f(x) = \frac{1}{b-a}$$ for $$a \leq x \leq b
Step 6:: Interpret the Probability
This means for any two sub-intervals of equal width within $$[a,b]$$, the probability of the random variable falling in either interval is exactly the same.
Final Answer
A uniform random variable is just as likely to assume a value in one interval as it is to assume a value in any other interval of equal size.
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