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Uniform(0,1) and Bernoulli(1/4)

Source: LIMIT 2019 CCS2 P5

April 28, 2021
probability

Problem Statement

Suppose that X∼Uniform⁡(0,1)X\sim\operatorname{Uniform}(0,1) and Y∼Bernoulli⁡(14)Y\sim\operatorname{Bernoulli}\left(\frac14\right), independently of each other. Let Z=X+YZ=X+Y. Then which of the following is true? <spanclass=′latex−bold′>(A)</span> The distribution of Z is symmetric about 1<span class='latex-bold'>(A)</span>~\text{The distribution of }Z\text{ is symmetric about }1 <spanclass=′latex−bold′>(B)</span> Z has a probability density function<span class='latex-bold'>(B)</span>~Z\text{ has a probability density function} <spanclass=′latex−bold′>(C)</span> E(Z)=54<span class='latex-bold'>(C)</span>~E(Z)=\frac54 <spanclass=′latex−bold′>(D)</span> P(Z≤1)=14<span class='latex-bold'>(D)</span>~P(Z\le1)=\frac14
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