Mathematics · Probability and Statistics
Let X be a random variable with probability density function f(x) = 1π (1 + x2), x ∈ &mathbb;R. If Z = 12 + 1π tan-1(X), then which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1E(Zm) = 1m + 1 for all m ∈ &mathbb;N✓
- 2E(Φ-1(Z)) = 0, where Φ(·) is the cumulative distribution function of a standard normal random variable.
- 3Z is degenerate at 0.
- 4If Z1 and Z2 are i.i.d. random variables with the same distribution as Z, then Z = Z1 + Z22.
Correct answer
E(Zm) = 1m + 1 for all m ∈ &mathbb;N
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