MathDB

Problem 5

Part of 2019 LIMIT Category C

Problems(2)

nontrivial subgroups, finitude or infinitude

Source: LIMIT 2019 CCS1 P5

4/28/2021
Let G=(S1,⋅)G=(S^1,\cdot) be a group. Then its nontrivial subgroups <spanclass=′latex−bold′>(A)</span> are necessarily finite<span class='latex-bold'>(A)</span>~\text{are necessarily finite} <spanclass=′latex−bold′>(B)</span> can be infinite<span class='latex-bold'>(B)</span>~\text{can be infinite} <spanclass=′latex−bold′>(C)</span> can be dense in S1<span class='latex-bold'>(C)</span>~\text{can be dense in }S^1 <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
group theoryabstract algebra
Uniform(0,1) and Bernoulli(1/4)

Source: LIMIT 2019 CCS2 P5

4/28/2021
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
probability