True or False:
Let $X$ be a Random variable.Then the function $G(t):=P(X \leq t)+\frac{P(X \geq t)}{2}$ is right-continuous.
Any ideas?
Cumulative distribution function and continuity
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real-analysis
probability
probability-theory
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0Note that $ G(t) = F_X(t) + \frac{1-F_X(t)}{2} + \frac{P(X=t)}{2} $. Is $ P(X=t) $ right continuous for every random variable $ X$? – 2017-01-21
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0Your four last questions on the site are completely lacking of context. Are you happy with that? – 2017-01-22
1 Answers
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Consider the random variable $X$ with $$P(X=1)=1,\quad P(X\neq 1)=0.$$ Then for any $t>1$ we have $$P(X\leq t)=1,\quad P(X\geq t)=0,$$ and therefore $$\lim_{t\to 1^+}G(t)=1+\frac{0}{2}=1.$$ However $$G(1)=P(X\leq 1)+\frac{P(X\geq 1)}{2}=1+\frac{1}{2}=\frac{3}{2}.$$