Homework 1 - Random Distribution Related
1.3【 CDF & 各种期望值 】
To solve this problem, we need to calculate several statistical properties for the given discrete probability distribution:
- The expected value ⟨ x ⟩ \langle x \rangle ⟨x⟩ (mean)
- The expected value of x 2 x^2 x2, ⟨ x 2 ⟩ \langle x^2 \rangle ⟨x2⟩
- The standard deviation σ \sigma σ
- The variance σ 2 \sigma^2 σ2
- The cumulative distribution function (CDF)
Step-by-Step Solution:
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Define the probability distribution:
- P ( x = 0 ) = 1 2 P(x = 0) = \frac{1}{2} P(x=0)=21
- P ( x = 1 ) = 1 3 P(x = 1) = \frac{1}{3} P(x=1)=31
- P ( x = 2 ) = 1 6 P(x = 2) = \frac{1}{6} P(x=2)=61
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Calculate ⟨ x ⟩ \langle x \rangle ⟨x⟩:
⟨ x ⟩ = ∑ i x i P ( x i ) \langle x \rangle = \sum_{i} x_i P(x_i) ⟨x⟩=i∑xiP(xi) -
Calculate ⟨ x 2 ⟩ \langle x^2 \rangle ⟨x2⟩:
⟨ x 2 ⟩ = ∑ i x i 2 P ( x i ) \langle x^2 \rangle = \sum_{i} x_i^2 P(x_i) ⟨x2⟩=i∑xi2P(xi) -
Calculate the variance σ 2 \sigma^2 σ2:
σ 2 = ⟨ x 2 ⟩ − ⟨ x ⟩ 2 \sigma^2 = \langle x^2 \rangle - \langle x \rangle^2 σ2=⟨x2⟩−⟨x⟩2 -
Calculate the standard deviation σ \sigma σ:
σ = σ 2 \sigma = \sqrt{\sigma^2} σ=σ2 -
Calculate the CDF:
The CDF F ( x ) F(x) F(x) is the sum of the probabilities up to and including x x x.
The calculations for the given discrete probability distribution are as follows:
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Expected value ⟨ x ⟩ \langle x \rangle ⟨x⟩:
⟨ x ⟩ = 0 ⋅ 1 2 + 1 ⋅ 1 3 + 2 ⋅ 1 6 = 0 + 1 3 + 1 3 = 2 3 \langle x \rangle = 0 \cdot \frac{1}{2} + 1 \cdot \frac{1}{3} + 2 \cdot \frac{1}{6} = 0 + \frac{1}{3} + \frac{1}{3} = \frac{2}{3} ⟨x⟩=0⋅21+1⋅31+2⋅61=0+31+31=32 -
Expected value of x 2 x^2 x2, ⟨ x 2 ⟩ \langle x^2 \rangle ⟨x2⟩:
⟨ x 2 ⟩ = 0 2 ⋅ 1 2 + 1 2 ⋅ 1 3 + 2 2 ⋅ 1 6 = 0 + 1 3 + 4 6 = 1 3 + 2 3 = 1 \langle x^2 \rangle = 0^2 \cdot \frac{1}{2} + 1^2 \cdot \frac{1}{3} + 2^2 \cdot \frac{1}{6} = 0 + \frac{1}{3} + \frac{4}{6} = \frac{1}{3} + \frac{2}{3} = 1 ⟨x2⟩=02⋅21+12⋅31+22⋅61=0+31+64=31+32=1</