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Suppose that a random variable $X$ has a probability density function (pdf) given by
\[f(x)=c \exp (-2 x)\]
for $x \geq 1$, and $f(x)=0$, for $x<1$, where $c$ is an appropriate constant so that $f(x)$ is a valid pdf.

What is the expected value of $X$ given that $X \geq 5?$ 

  1. $5 \frac{1}{2}$
  2. $7$
  3. $10$
  4. $8 \frac{1}{2}$
  5. $6$
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