Grinstead-Snell-Probability · Problem 6.3.4

Sums of Random Variables

Mathematics · expected-value-variance-continuous

Problem

Let XX be a random variable with range [−1,1][-1, 1] and density function fX(x)=ax+bf_X(x) = ax + b if ∣x∣<1|x| < 1. (a) Show that if ∫−1+1fX(x) dx=1\int_{-1}^{+1} f_X(x)\,dx = 1, then b=1/2b = 1/2. (b) Show that if fX(x)≥0f_X(x) \geq 0, then −1/2≤a≤1/2-1/2 \leq a \leq 1/2. (c) Show that μ=(2/3)a\mu = (2/3)a, and hence that −1/3≤μ≤1/3-1/3 \leq \mu \leq 1/3. (d) Show that σ2(X)=(2/3)b−(4/9)a2=1/3−(4/9)a2\sigma^2(X) = (2/3)b - (4/9)a^2 = 1/3 - (4/9)a^2.

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