Question

3. Let X have pdf $f(x) = cx(1 - x)$ for $0 < x < 1$. (a) Find the constant c so that $f(x)$ is a pdf, and hence, find the cdf of X (b) Find the pdf of $Y = X^2$ (c) Derive $E(X^2)$ using the pdf of X, and $E(Y)$ using the pdf of Y. Compare. (d) Derive $E(Y^2)$ by any method, and hence, find the variance of Y.

          3. Let X have pdf $f(x) = cx(1 - x)$ for $0 < x < 1$.
(a) Find the constant c so that $f(x)$ is a pdf, and hence, find the cdf of X
(b) Find the pdf of $Y = X^2$
(c) Derive $E(X^2)$ using the pdf of X, and $E(Y)$ using the pdf of Y. Compare.
(d) Derive $E(Y^2)$ by any method, and hence, find the variance of Y.

3. Let X have pdf f(x) = cx(1 - x) for 0 < x < 1.
(a) Find the constant c so that f(x) is a pdf, and hence, find the cdf of X
(b) Find the pdf of Y = X^2
(c) Derive E(X^2) using the pdf of X, and E(Y) using the pdf of Y. Compare.
(d) Derive E(Y^2) by any method, and hence, find the variance of Y.

Added by Victoria B.

Principles of Economics

Gregory Mankiw 8th Edition

Instant Answer

Solved by Expert Niamat Khuda

Step 1

∫[0,1] cx(1-x) dx = 1 To solve this integral, we can expand the expression and integrate term by term: c∫[0,1] (x - x^2) dx = 1 c(∫[0,1] x dx - ∫[0,1] x^2 dx) = 1 c(1/2 - 1/3) = 1 c(3/6 - 2/6) = 1 c(1/6) = 1 c = 6 So the constant c is 6. Now, to find Show more…

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Let X have pdf f = cx(1-x) for 0 < x < 1. a) Find the constant c so that f is a pdf, and hence, find the cdf of X. b) Find the pdf of Y = X^2. c) Derive E(X) using the pdf of X and E(Y) using the pdf of Y. Compare. d) Derive E(Y^2) by any method, and hence, find the variance of Y.