Functions of Random Variable.

#1
Let X be exponential(lambda), and let Y=max(1,X).
Find the cdf of Y. Also sketch the cdf.
Suppose that X is discrete with pmf p(0)=p(1)=2p(2) (and zero otherwise).
Find the pmf and cdf of X.
How would you simulate the random variable X starting with U, a uniform[0,1] random variable?
That is, find a function g, such that g(U) has the same pmf/cdf as X.
Mod Note: Please don't double post the same question.
 
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BGM

TS Contributor
#2
Note \( Y \) has a mass on 1 and continuous above.

It is because
\( Y = \max\{1, X\} = \left\{\begin{matrix} 1 & \mathrm{if} & X \leq 1 \\
X & \mathrm{if} & X > 1 \end{matrix}\right. \)

Therefore
\( F_Y(y) = \Pr\{Y \leq y\} = \left\{\begin{matrix}
0 & \mathrm{if} & y < 1 \\
F_X(y) & \mathrm{if} & y \geq 1 \end{matrix}\right. \)

For the second question, use \( p(0) + p(1) + p(2) = 1 \) with the given constraint.

I guess for the simulation part it is not hard. Ask again if you still have questions.