PDE ■P(a≤X≤a)=∫%fx(x)dx=0. P(a≤X≤b)=P(a<X≤b) =P(a≤X<b)=P(a<X<b) The entire area under the graph is equal to 1. 00 fx(x)dx=P(-oo≤X≤o)=1 -00 PDF fx(a) Sample space Event{a≤Xsb)
PDF 𝑃 𝑎 ≤ 𝑋 ≤ 𝑎 = 𝑓𝑋 𝑥 𝑑𝑥 𝑎 𝑎 = 0. ∴ 𝑃 𝑎 ≤ 𝑋 ≤ 𝑏 = 𝑃 𝑎 < 𝑋 ≤ 𝑏 = 𝑃 𝑎 ≤ 𝑋 < 𝑏 = 𝑃 𝑎 < 𝑋 < 𝑏 The entire area under the graph is equal to 1. 𝑓𝑋 𝑥 𝑑𝑥 ∞ −∞ = 𝑃 −∞ ≤ 𝑋 ≤ ∞ = 1
Interpretation of PDF ·fx(x):“probability mass per unit length” ■P(lx,x+D=+fx()at≈fr(x)G PDF fx(x) xE+δ
Interpretation of PDF 𝑓𝑋(𝑥): “probability mass per unit length” 𝑃 𝑥, 𝑥 + 𝛿 = 𝑓𝑋 𝑡 𝑑𝑡 𝑥+𝛿 𝑥 ≈ 𝑓𝑋(𝑥) ∙ 𝛿
Example 1:Uniform Consider a random variable X takes value in interval a,b]. Any subintervals of the same length have the same probability. It is called uniform random variable
Example 1: Uniform Consider a random variable 𝑋 takes value in interval 𝑎, 𝑏 . Any subintervals of the same length have the same probability. It is called uniform random variable
Example 1:Uniform Its PDF has the form ifa≤x≤b otherwise PDF fx(x) 1 b-a 0 b
Example 1: Uniform Its PDF has the form 𝑓𝑋 𝑥 = 1 𝑏 − 𝑎 , if 𝑎 ≤ 𝑥 ≤ 𝑏 0, otherwise
Example 2:Piecewise Constant When sunny,driving time is 15-20 minutes. When rainy,driving time is 20-25 minutes. With all times equally likely in each case. Sunny with prob.2/3,rainy with prob.1/3 The PDF of driving time X is C1, if15≤x≤20 fx(x)= C2, if20≤x≤25 0, otherwise
Example 2: Piecewise Constant When sunny, driving time is 15-20 minutes. When rainy, driving time is 20-25 minutes. With all times equally likely in each case. Sunny with prob. 2/3, rainy with prob. 1/3 The PDF of driving time 𝑋 is 𝑓𝑋 𝑥 = 𝑐1, if 15 ≤ 𝑥 ≤ 20 𝑐2, if 20 ≤ 𝑥 ≤ 25 0, otherwise