Answer to Question #183630 in Statistics and Probability for Robert

Question #183630

For a continuous function f(x) =kx2e-x where x>0.Find i) k, ii) mean, iii) variance

1
Expert's answer
2021-05-04T13:05:31-0400

f(x)=kx2ex,  x>0f(x) =kx^2e^{-x}, \; x>0


i) f(x)dx=1\int f(x) dx =1

0kx2exdx=k0x2exdx=/integrating by parts/=2k=1\int_0^\infty kx^2 e^{-x}dx = k \int_0^\infty x^2 e^{-x} dx =/ \text{integrating by parts}/ = 2k =1

k=1/2k=1/2


ii) mean

M[x]=0xf(x)dx=k0x3exdx=/integrating by parts/=6k=3M[x] = \int_0^\infty xf(x) dx= k\int_0^\infty x^3 e^{-x} dx = / \text{integrating by parts}/ = 6k =3

M[x]=3M[x] = 3


iii) variance

D[x]=M[x2](M[x])2D[x] = M[x^2] - (M[x])^2

M[x2]=0x2f(x)dx=k0x4exdx=/integrating by parts/=24k=12M[x^2] = \int_0^\infty x^2f(x) dx= k\int_0^\infty x^4 e^{-x} dx = / \text{integrating by parts}/ = 24k =12

D[x]=12(3)2=129=3D[x] = 12 - (3)^2 = 12 - 9 = 3


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