STAT/MATH 309 DISCUSSION 5
TA: Jingjiang(Jack) Peng
Office: 1275 MSC, 1300 Universtiy Avenue
E-mail: [email protected]
Phone: 262-1577
Office Hour: 11:30-1:30 p.m. Tuesday or by appoitment
Websit: www.stat.wisc.edu/∼peng
1 Cumulative Distribution Function
•c.d.f FX(x) = P(X≤x)
•(i)0 ≤FX(x)≤1;(ii) FX(∞) = 1, and FX(−∞) = 0; (iii) FX(x) is non-decreasing
•For discrete random variables, FX(x) = P(X≤x) = Pxi≤xP(X=xi)
•For continuous random variable, FX(x) = Rx
−∞ fX(t)dt, Where fXis called probability
density function
2 One-dimensional Change of Variable
•If X is discrete random variable, Y=h(X), then pY(y) = P(Y=y) = P(h(X) =
x) = P(X=h−1(x)) = Px∈h−1(y)P(X=x)
•If X is continous, Y=h(X), then fY(y) = fX(h−1(y)
|h′(h−1(y))|
3 Examples
1: (2.5.4) Let X∼N(0,1), calculate P(X≤ −5), P (−2≤X≤7), P (X≥3)
2: (2.5.5) Y∼N(−8,4), calculate P(Y≤ −5), P (−2≤Y≤7), P (X≥3)
3: (2.6.3) X∼N(µ, σ2), Y =cX +d, where c > 0. Prove Y∼N(cµ +d, c2σ2)
4: (2.6.5) X∼exp(λ), Y=X3, Compute the density fYof Y.
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