please help me to understand fully following definition :

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i am using this book


page 79,i can't understand some part,in spite of this fact that on next page there are explanations of these,for example part C,does it means that as $h$ approaches 0 then distribution function converges to actual function?then why not it is continuous from left?about non decreasing,i have read that

follows from ${X < x } \subset {X < y }$ for $x < y$

for part $b$

$ P[{X < -n}] -> 0$ and $P[{X < n}] -> 1.$

what does this part means?thanks in advance


1 Answer 1


For C. it only means that for every $x_0\in\mathbb{R}$, $$ F_X(x_0+h) \xrightarrow[h\to0^+]{} F_X(x_0) $$ Why not from the left? Consider a random variable $X$ which has probability $1$ of having taking value $78$ (for instance). That is, $X$ is almost surely equal to $78$, there is not much randomness there... then, $$ F_X(x) = \begin{cases} 0 & \text{if } x < 78\\ 1 & \text{if } x \geq 78\\ \end{cases} $$ which is certainly not left-continuous (but is right-continuous).

For non-decreasing, well: the probability that $X \leq 10$ is definitely not more than the probability that $X \leq 11$ (since if $X \leq 10$, you also have $X \leq 11$). This holds for any $a\leq b$ instead of 10 and 11, and is equivalent to saying $F_X$ is non-decreasing (by definition of $F_X(x)=\mathbb{P}\{X\leq x\}$).

Finally, when $n$ goes to $-\infty$, the probability that $\mathbb{P}\{X\leq n\}$ does go to $0$ (the smaller the value of $n$, the smaller the probability that the random value taken by $X$ will be below $n$).

  • $\begingroup$ thanks for help,i have understood everything $\endgroup$ Commented May 13, 2014 at 19:44

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