Expected Value of exponential funtion...

Generally speaking, the mgf of the normal random variable X with mean u and standard deviation sigma is






圖片參考：http://upload.wikimedia.org/math/e/9/6/e96e7b41e29c972e236678ec06fb546d.png


圖片參考：http://upload.wikimedia.org/math/3/b/f/3bfacbb6d1055326935bcea17f8f3973.png




圖片參考：http://upload.wikimedia.org/math/5/9/f/59f6e3d8bcd23929b436c0bf33ac9afa.png





圖片參考：http://upload.wikimedia.org/math/0/0/e/00e3bfb259ad5c9a6f800dbcb5c784c4.png



Now you can substitute to get
E[exp(X)]=exp( μ+σ^2/2)
For normal random variable Z with u=0 and sigma =1
E[exp(Z)]=exp( 0+1/2)=exp(1/2)
What is the relationship between Normal R. V and exponential function??
無relationship 的﹐你問的其實是mgf問題。

2007-04-23 03:56:07 補充：
應是Now you can substitute t=1 to get