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Coursework questions

1.
The lifetime (in years) of a light bulb $Y$ has a continuous distribution with probability density function

$f_{Y}(y)=\left\{\begin{array}[]{cl}y\exp(-y^{2}/2),&y>0,\\ 0&\mbox{otherwise.}\end{array}\right.$
1. (a)
  
  For a fixed $y>0$ , find the probability the bulb survives at least time $y$ .
2. (b)
  
  Find the probability that the bulb’s lifetime is between one and two years.
[2]
2.
1. (a)
  
  Find the median $x_{0.5}$ of the random variable $X\sim{\rm Exp}(\lambda)$ (i.e. with pdf $f_{X}(x)=\lambda\exp(-\lambda x)$ for $x>0$ and $f_{X}(x)=0$ otherwise).
2. (b)
  
  Find $x_{0.25}$ and $x_{0.75}$ when $X$ has probability density function $f_{X}$ given by
  
  $f_{X}(x)=\left\{\begin{array}[]{cl}8/(x+2)^{3},&x>0,\\ 0&\mbox{otherwise.}\end{array}\right.$
[4]
3.
Let $X\sim{\rm U}(0,1)$ be a uniformly distributed random variable with parameters 0 and 1. Let $Z=-\log(1-X)$ , so that $Z$ is a random variable taking values in $(0,\infty)$ .
1. (a)
  
  For arbitrary $z>0$ , find $P(Z\leq z).$
2. (b)
  
  Find the probability density function of $Z$ . What is the distribution of $Z$ ?
[4]