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題組內容

四、Y₁ and Y₂ are two continuous random variables. Let Y₁ and Y₂ have the joint probability density function given by

(d) Please find the expected value of Y₁, E(Y₁).

其他申論題

(c) Please derive the factor demand function for labor (L) and capital (K). (10 分)

(a) Find the value of k that makes this a probability density function. (5 分)

(c) Find the marginal density function for Y1 and Y2.

(a) (10%) Please give an ε and δ and show that .(Hint: ||a|-|b||＜|a-b|)

(b) (10%) Let g be a function defined at least on some open interval containing the number 0. We say that g is of smaller order than h, or that g(h) is little-o(h) and write g(h) = o(h), if and only if g(h) is small enough compared with h that = 0. Show that h3 = o(h).

2. (15%) Show that ln(1+x) ≈ x as x→0. (Hint: (ln(x))'=1/x)

(a) (10%) Find the critical numbers of F and determine theintervals on which F is increasing and the intervals onwhich F is decreasing.

(b) (10%) Determine the concavity of the graph of F and find thepoints of inflection (if any).

(a) (10%) Verify that the region below the graph of f(x) = 1/x, x≥1 has infinite area.

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