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Unformatted text preview: 4.3 Expected Values For Continuous Random Variables (CRV) Definition: The expected value for a continuous random variable Y is E (Y) where m = E (Y ) = ò ¥ ¥ yf ( y ) dy Theorem: Let g (y) be a function of c.r.v. Y then E ( g ( y )) =
Theorems: ò ¥ ¥ g ( y ) f ( y ) dy If c is any constant and Y a c.r.v. then 1. E(c) = c 2. E[c g(y)] = c E( g(y) ) 3. E[g 1(y) + g 2(y) +…..g k(y)] = E[g 1(y)] +……+E[g k(y)] Example: #4.15 Pg. 164 If Y has pdf f(y) = (3/2) y2 + y , 0 , 0<y<1 other wise Find µ and σ2 Solution: Recall σ2 = V(Y) = E(Y2) – [E(Y )]2 = E(Y2) – µ2 Now
m = E (Y ) =
¥ ¥ 1 ò yf ( y)dy
æ3
0 = òç2 è ö y 3 + y 2 ÷ dy ø
1 3 1 = y4 + y3 8 3 =
0 3 1 17 += 8 3 24 E (Y ) = 2 ∞ y2 f(y) dy ∞
1 = 0 = 3 y5 + 1 y4 10 4 3 y4 + y3 dy 2
1 0 = 3 + 1 = 6 + 5 = 11 10 4 20 20 20
2 ∴ σ2 = E (Y2) – µ2 = 11 – 17 20 24 = 0.048 Hw pg. 164 # 4.14, 4.16, 4.19, 4.26 ...
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This note was uploaded on 01/30/2011 for the course MAT 219 taught by Professor Ab during the Spring '10 term at Coast Guard Academy.
 Spring '10
 AB

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