2. Assume that the pdf of the random variable x is uniform in the interval (10,...

Question

Question

2. Assume that the pdf of the random variable x is uniform in the interval (10, 12) and y = x^3.

(a) Find fy (y).

(b) Find E{y}.

math Statistics-And-Probability

Answer 1

Answer #1

a)

here CDF of Y: F(y)=P(Y<y)=P(X³ <y)=P(X<y^1/3)= $\int_{10}^{y^{1/3}}$ (f(x) dx= $\int_{10}^{y^{1/3}}$ (1/2) dx=(x/2)|^y^1/3₁₀ =(1/2)*(y^1/3-10^1/3)

therefore pdf f(y)=(d/dy)*F(y)=(1/6)*y^-2/3 for 1000<y<1728

b)

E(Y)=E(X³)= $\int_{10}^{{12}$ x³(f(x) dx = $\int_{10}^{{12}$ x³/2 dx =x⁴/8 |¹²₁₀ =1342

Answer 2

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