By Fearn T., Brown P.J., Besbeas P.

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5 6(1 − P (A) = P (A ∩ R) = 0 1 β dβ = . 5 6(1 − P (A) = P (A ∩ R) = 0 α/β) dβ d α. α This also provides the result—at the expense of a more difficult integration. 6. cls 48 QC: IML/FFX T1: IML October 27, 2006 7:23 INTERMEDIATE PROBABILITY THEORY FOR BIOMEDICAL ENGINEERS (b) For 0 < α < 1, 1 f x (α) = √ α/β) dβ = 6(1 − 2 α + α). 6(1 − α (c) For 0 < β < 1, β f y (β) = 6(1 − α/β) d α = 2β. , 0 ≤ α ≤ β ≤ 1), β α Fx,y (α, β) = 6 (1 − α /β ) dβ d α 0 α α =6 (β − 2 α β + α ) d α 0 √ = 6αβ − 8α αβ + 3α 2 .

A) Write a mathematical expression for f x (α). (b) Determine E(x). (c) Suppose grades are assigned on the basis of: 90–100 = A = 4 honor points, 75–90 = B = 3 honor points, 60–75 = C = 2 honor points, 55–60 = D = 1 honor point, and 0–55 = F = 0 honor points. Find the honor points PDF. (d) Find the honor points average. 8. 5) + δ(α) + δ(α − 2). 2 8 8 Determine: (a) E(x), (b) σx2 . 3: Probability density function for Problem 7. 9. A PDF is given by 2 3 1 1 f x (α) = δ(α + 1) + δ(α) + δ(α − 1) + δ(α − 2).

Let random variable x equal the number of students who earn an A in the class. Determine: (a) p x (α), (b) E(x), (c) σx . 4. 5(α + 1), 0, −1 < α < 1 otherwise. 2 Determine: (a) E(x), (b) σx2 , (c) E(1/(x + 1)), (d) σ1/(x+1) . 5. The PDF for random variable y is f y (yo ) = sin(yo ), 0, 0 < yo < π/2 otherwise, and g (y) = sin(y). Determine E(g (y)). 6. 5e −|α| , (b) f x (α) = 5e −10|α| . 7. The grade distribution for Professor S. Rensselaer’s class in probability theory is shown in Fig. 3. (a) Write a mathematical expression for f x (α).