# Functions of Random Variable.

#### Actuarial_Deepika

##### New Member
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
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.