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QUESTION
 A random variable X has age specific failure rate function
  $\lambda (x)=x$. Find f(x) and F(x) and derive the mode and
  median of the distribution of x.

ANSWER $G(x)=e^{-\int\lambda(x)\,dx}=e^{-\int
x\,dx}=e^{-\frac{1}{2}x^2}\\
  f(x)=\lambda(x)G(x)=xe^{-\frac{1}{2}x^2}\\
  F(x)=1-G(x)=1-e^{-\frac{1}{2}x^2}$\\
  Mode m: $f'(m)=0, f'(x)=e^{-\frac{1}{2}x^2}$ therefore m=1\\
  Median M: $F(M)=G(M)=\frac{1}{2},\ \
  e^{-\frac{1}{2}M^2}=\frac{1}{2},\ \ \frac{1}{2}M^2=\ln 2,\ \
  \\M=\sqrt{2 \ln 2}=1.177>m$

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