Question

Introductory Econometrics, please help!

Question 2). (27.5) Assume the following p.d.f f(x) can be used to describe the Income distribution in the US f(x) -kx'2 for 0sx<6 (a) Find the value of k Now, assume the following pdf: f(x) = 1/72x2 for 0Ex6 (b) sketch a graph of the corresponding pdf. (c) sketch a graph of the corresponding cdf. (d) Find P(2s X s 4) and indicate what you calculate in your graph in Part b) (5) (e) Find the expected value ofx. Explain the meaning of the expected value in this particular example. 2 sentences Find the variance of x. Explain the meaning of the variance in this particular example. 2 sentences.

Answer 1

f(x) = kx^2, 0 lessthanorequalto x lessthanorequalto 6, a) for a legitimate pdf, integral f(x) dx = 1 k [x^3/3]_0^6 = 1 thus k [36 times 6/3] = 1 thus k = 1/72 Am Now f(x) = x^2/72 b) Graph, f(0) = 0, f(6) = .5 c) cdf, c df f(x) = integral f(x) dx = x^3/3 times 72 = x^3/216 f(x) = x^3/216 d) P[2 lessthanorequalto x lessthanorequalto 4] = > f(4) - f(2) = (4)^3/216 - (2)^3/216 = 64 - 8/216 = 56/216 = .259 Am e) E(x) = integral x f(x) dx = integral x^3/72 dx = x^4/72 middot 4 = x^4/288 Am expected value means avg value of x. f) var (x) = E(x^2) - {E(x)}^2 E(x^2) = integral x^2 f(x) dx = integral x^4/72 dx = x^5/5 middot 72 = x^5/360 so var (x) = x^5/360 - (x^4/288)^2