Tutorial 8:Further Topics on Random variables 1 Weiwen LIU wwliu@cuhk.edu.hk March 20,2017 1
Tutorial 8: Further Topics on Random Variables 1 Weiwen LIU wwliu@cuhk.edu.hk March 20, 2017 1
Derived distributions Given Y=g(X)of a continuous random variable X and PDF of X, how to calculate the PDF of y? ·Two step approach 2
Derived distributions • Given 𝑌 = 𝑔(𝑋) of a continuous random variable 𝑋 and PDF of 𝑋, how to calculate the PDF of 𝑌? • Two step approach 2
Calculation of PDF of Y =g() 1.Calculate the CDF Fy of Y using the formula Fy)=P(g(X)≤y)= fx(x)dx Jxlg(x)≤y} 2.Differentiate Fy to obtain the PDF fy of Y: dFy fy(y)=dy (y). 3
Calculation of PDF of 𝑌 = 𝑔(𝑋) 1. Calculate the CDF 𝐹𝑌 of 𝑌 using the formula 𝐹𝑌 𝑦 = 𝑃 𝑔 𝑋 ≤ 𝑦 = න {𝑥|𝑔(𝑥)≤𝑦} 𝑓𝑋(𝑥)𝑑𝑥 2. Differentiate 𝐹𝑌 to obtain the PDF 𝑓𝑌 of 𝑌: 𝑓𝑌 𝑦 = 𝑑𝐹𝑌 𝑑𝑦 𝑦 . 3
Calculation of PDF of Y =g() 。Linear case: The PDF of Y ax +b in terms of the PDF X. 0-票)=奇。) 。Monotonic Case: Suppose that g is monotonic and that for some function h and all x in the range of X we have y=g(x)if and only if x =h(y). Assume that h is differentiable. i0)=6o=例 4
Calculation of PDF of 𝑌 = 𝑔(𝑋) • Linear case: • The PDF of 𝑌 = 𝑎𝑋 + 𝑏 in terms of the PDF 𝑋. 𝑓𝑌 𝑦 = 𝑑𝐹𝑌 𝑑𝑦 𝑦 = 1 |𝑎| 𝑓𝑋 𝑦 − 𝑏 𝑎 • Monotonic Case: • Suppose that 𝑔 is monotonic and that for some function ℎ and all 𝑥 in the range of 𝑋 we have 𝑦 = 𝑔 𝑥 if and only if 𝑥 = ℎ(𝑦). • Assume that ℎ is differentiable. 𝑓𝑌 𝑦 = 𝑓𝑋 ℎ(𝑦) 𝑑ℎ 𝑑𝑦 (𝑦) 4
Example 1 Let X be a random variable with PDF fx.Find the PDF of the random variable y =X ·(a)when fx(x)= f-2<x≤1, 0, otherwise; (b)when fx()=2e otherwise; .(c)for general fx(x). 5
Example 1 • Let X be a random variable with PDF 𝑓𝑋. Find the PDF of the random variable 𝑌 = |𝑋|, • (a) when 𝑓𝑋 𝑥 = ቐ 1 3 , 𝑖𝑓 − 2 < 𝑥 ≤ 1, 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒; • (b) when 𝑓𝑋 𝑥 = ቊ 2𝑒 −2𝑥 , 𝑖𝑓 𝑥 > 0, 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒; • (c) for general 𝑓𝑋(𝑥). 5