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I have the following density function:

$f(t)=\begin{cases} 0.8\exp(-2t)+0.5 \sqrt\frac{2}{\pi}\exp(-\frac{t^2}{2}) \text{, if } t > 2 \\0.8\exp(-2t)+0.5 \sqrt\frac{2}{\pi}\exp(-\frac{t^2}{2})+0.05 \text{, if } 0 \leq t \leq 2 \\ 0, \text{otherwise} \end{cases}$

How can I use a uniform distribution and possibly others such as (exponential and normal) to generate samples from this distribution?

If $t$ was finite, I could spit up the $t$ axis into chunks, then use a uniform random variable generator and take those values as CDF values and map it to $t$.

Not sure how to start here. Can we consider a probability of $0.5$ that $t$ is either less than $2$ or greater than $2$?

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Your probability density function is a linear combination of pdf's of the following three random variables with weights $0{.}5$, $0{.}4$ and $0{.}1$ correspondingly:

1) absolute value of standard normal $|X|$, $X\sim \mathcal N(0,1)$,

2) exponential $Y$ with mean $\frac12$,

3) uniform $Z$ on the interval $(0,\,2)$.

So you can simulate random choice with probabililties $0{.}5$, $0{.}4$ and $0{.}1$ of the simulated value of these random variables.

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