Probability | Part IA, 2005

Suppose c1c \geqslant 1 and XcX_{c} is a positive real-valued random variable with probability density

fc(t)=Actc1etc,f_{c}(t)=A_{c} t^{c-1} e^{-t^{c}},

for t>0t>0, where AcA_{c} is a constant.

Find the constant AcA_{c} and show that, if c>1c>1 and s,t>0s, t>0,

P[Xcs+tXct]<P[Xcs]\mathbb{P}\left[X_{c} \geqslant s+t \mid X_{c} \geqslant t\right]<\mathbb{P}\left[X_{c} \geqslant s\right]

[You may assume the inequality (1+x)c>1+xc(1+x)^{c}>1+x^{c} for all x>0,c>1x>0, c>1.]

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