Grinstead-Snell-Probability · Problem 6.3.5

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)=ax2+bx+cf_X(x) = ax^2 + bx + c if ∣x∣<1|x| < 1 and 00 otherwise. (a) Show that 2a/3+2c=12a/3 + 2c = 1 (see Exercise 4). (b) Show that 2b/3=μ(X)2b/3 = \mu(X). (c) Show that 2a/5+2c/3=σ2(X)2a/5 + 2c/3 = \sigma^2(X). (d) Find aa, bb, and cc if μ(X)=0\mu(X) = 0, σ2(X)=1/15\sigma^2(X) = 1/15, and sketch the graph of fXf_X. (e) Find aa, bb, and cc if μ(X)=0\mu(X) = 0, σ2(X)=1/2\sigma^2(X) = 1/2, and sketch the graph of fXf_X.

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