Distribution of a random variable that is produced by another uniformly distributed random variable Assume a simple two-step lottery: the reward (denoted by $Y$) is a random variable and realized as follows:
First: random variable $X$ is drawn from uniform distribution by interval $(a,c)$.
Second: After the realization of $X$, a player determines reward ($Y$) stochastically:

*

*with probability $\alpha$: reward is $bX$  ($b$ is constant)


*with probability $1-\alpha$: reward is $X$
Now, what is the distribution of the random reward $Y$?
If the solution is weighted average on both states: i.e. $Y=\alpha bX+(1-\alpha)X=(\alpha b+1-\alpha )X$
then we can say $Y$ is a linear rescaling transformation of $X$ and so $Y$ has the same distribution as $X$ has. But I am suspicious that $Y$ really is a weighted average of $X$ and $bX$

The statement "$Y$ is a weighted average of $X$ and $bX$" is false.
What is true is: "the density (or CDF) of $Y$ is a weighted average of the densities (or CDF) of $X$ and $bX$".
This is known as a mixture.
