Math  /  Data & Statistics

Question3. 21. Supozojmë se XX ka densitet f(x)=ex,x>0f(x)=e^{-x}, \quad x>0 (a) Llogaritni funksionin prodhues të momenteve të XX dhe gjeni pritjen matematike, mesataren dhe dispersionin. (b) Gjeni pritjen matematike drejtpërsëdrejti nga përkufizimi.

Studdy Solution
Calculate the expectation directly from the definition:
E[X]=0xf(x)dx=0xexdx \mathbb{E}[X] = \int_{0}^{\infty} x f(x) \, dx = \int_{0}^{\infty} x e^{-x} \, dx
Use integration by parts, let u=x u = x and dv=exdx dv = e^{-x} \, dx , then du=dx du = dx and v=ex v = -e^{-x} :
E[X]=[xex]0+0exdx \mathbb{E}[X] = \left[ -x e^{-x} \right]_{0}^{\infty} + \int_{0}^{\infty} e^{-x} \, dx
=0+[ex]0=0+1=1 = 0 + \left[ -e^{-x} \right]_{0}^{\infty} = 0 + 1 = 1
The expectation is E[X]=1 \mathbb{E}[X] = 1 , the variance is Var(X)=1 \text{Var}(X) = 1 .

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