✔ 最佳答案
1) Moment generating function of X:
MX(t) = eμt+(σ^2t^2)/2
For Z = X/ σ - μ/σ, its MGF is:
MZ(t) = e-μt/σ MX(t/σ)
= e-μt/σ eμt/σ+(t^2)/2
= e(t^2)/2
which is an MGF of normal distribution with mean = 0 and standard deviation = 1
2) The Beta function is given by:
2∫(x = 0 → π/2) sin3 x cos17 x dx
= 2∫(x = π/2 → 0) sin2 x cos17 x d(cos x)
= 2∫(x = π/2 → 0) cos17 x - cos19 x d(cos x)
= 2[(cos18 x)/18 - (cos20 x)/20] (x = π/2 → 0)
= 1/90
So the pdf of the beta distribution is:
f(x) = 90x(1 - x)8
And the required probability is:
∫(x = 0 → 0.1) 90x(1 - x)8 dx
= 90∫(x = 0 → 0.1) (x - 1)(1 - x)8 + (1 - x)8 dx
= 90∫(x = 0 → 0.1) [(1 - x)8 - (1 - x)9] dx
= 90∫(x = 0.1 → 0) [(1 - x)8 - (1 - x)9] d(1 - x)
= 90 [(1 - x)9/9 - (1 - x)10/10] (x = 0.1 → 0)
= 0.2643
