Math Problem Statement
Let Z ∼ N(0, 1). Find E(Φ(Z)), where Φ(·) is the cumulative distribution function of Z.
Solution
Let , meaning follows the standard normal distribution with mean and variance . We are tasked with finding the expected value , where is the cumulative distribution function (CDF) of a standard normal random variable .
Step-by-Step Solution
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Understanding :
- The function represents the probability that a standard normal variable is less than or equal to , i.e., where .
- Since , the problem asks for the expected value of the CDF of a standard normal random variable evaluated at another standard normal random variable.
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Symmetry of the Standard Normal Distribution:
- The standard normal distribution is symmetric around 0. The expected value of , i.e., , is 0. This symmetry plays a key role in simplifying the problem.
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The Nature of :
- The function gives values between and , because it represents cumulative probabilities.
- The probability for a normally distributed variable is uniformly distributed in the interval . This result comes from the probability integral transform theorem.
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Expected Value of a Uniform Distribution:
- The expected value of a uniform random variable on is . Therefore, the expected value of is:
Conclusion:
Would you like further clarification or additional details on this? Here are 5 related questions for deeper exploration:
- What is the intuition behind the probability integral transform theorem?
- How does the symmetry of the normal distribution impact various calculations?
- What is the interpretation of , the inverse CDF of the standard normal distribution?
- How would the expected value change if instead?
- Can you derive the CDF of a normal distribution from first principles?
Tip: The cumulative distribution function (CDF) of any random variable gives the probability that will take a value less than or equal to a specific value. It’s especially useful when working with probabilities across a distribution.
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Cumulative Distribution Function (CDF)
Standard Normal Distribution
Expected Value
Formulas
E(Φ(Z)) = 0.5
Theorems
Probability Integral Transform Theorem
Suitable Grade Level
Undergraduate - Probability and Statistics
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