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Probability Question?
Let X be a random variable over the integers Z={...−2,−1,0,1,2,...}.
Let P(X=0)=0 and for i≠0 let P(X=i)=1Zα|i|α where Zα=∑∞i=−∞1/|i|^α
For α=2:
What is E[X]?
What is var[X]?
What is std[X]?
1 Answer
- kbLv 78 years agoFavorite Answer
Clarifying the notation:
P(X = 0) = 0 and otherwise P(X = i) = 1/(Zα |i|^α), where
Zα = Σ(-∞ to ∞, nonzero) 1/|i|^α = 2 * Σ(i = 1 to ∞) 1/|i|^α.
So, E[X]
= Σ(-∞ to ∞) i * P(X = i)
= Σ(-∞ to -1) i * P(X = i) + 0 * P(X = 0) + Σ(1 to ∞) i * P(X = i)
= Σ(-∞ to -1) i * 1/(Zα |i|^α) + Σ(1 to ∞) i * 1/(Zα |i|^α)
= (1/Zα) * [Σ(-∞ to -1) i/|i|^α + Σ(1 to ∞) i/|i|^α]
= (1/Zα) * [Σ(k = 1 to ∞) -k/|-k|^α + Σ(i = 1 to ∞) i/|i|^α], letting k = -i
= (1/Zα) * [Σ(k = 1 to ∞) -k/|k|^α + Σ(i = 1 to ∞) i/|i|^α]
= (1/Zα) * [Σ(i = 1 to ∞) -i/|i|^α + Σ(i = 1 to ∞) i/|i|^α], dummy index change
= (1/Zα) Σ(i = 1 to ∞) [-i/|i|^α + i/|i|^α]
= 0.
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So, E[X^2]
= Σ(-∞ to ∞) i^2 * P(X = i)
= Σ(-∞ to -1) i^2 * P(X = i) + 0^2 * P(X = 0) + Σ(1 to ∞) i^2 * P(X = i)
= Σ(-∞ to -1) i^2 * 1/(Zα |i|^α) + Σ(1 to ∞) i^2 * 1/(Zα |i|^α)
= (1/Zα) * [Σ(-∞ to -1) i^2/|i|^α + Σ(1 to ∞) i^2/|i|^α]
= (1/Zα) * [Σ(k = 1 to ∞) (-k)^2/|-k|^α + Σ(i = 1 to ∞) i^2/|i|^α], letting k = -i
= (1/Zα) * [Σ(k = 1 to ∞) k^2/|k|^α + Σ(i = 1 to ∞) i^2/|i|^α]
= (1/Zα) * 2 Σ(i = 1 to ∞) i^2/|i|^α, dummy variable change
= (2/Zα) Σ(i = 1 to ∞) 1/|i|^(α-2).
If α = 2, then E[X^2] = (2/Z₂) Σ(i = 1 to ∞) 1 = ∞.
So, the variance and standard deviation equal ∞.
Note: We need α > 3 for the variance to be finite.
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I hope this helps!
