Cdf of a discrete random variable and convergence of distributions...by Artusartos Tags: convergence, discrete, distributions, random, variable 

#1
Jan2613, 03:28 PM

P: 245

In the page that I attached, it says "...while at the continuity points x of [itex]F_x[/itex] (i.e., [itex]x \not= 0[/itex]), [itex]lim F_{X_n}(x) = F_X(x)[/itex]." But we know that the graph of [itex]F_X(x)[/itex] is a straight line y=0, with only x=0 at y=1, right? But then all the points to the right of zero should not be equal to the limit of [itex]F_{X_n}(x)[/itex], right? Because [itex]F_X(x)[/itex] is always zero at those points, but [itex]F_X(x)[/itex] is 1? So how do I make sense of that?
Thanks in advance 



#2
Jan2713, 10:58 AM

Sci Advisor
P: 3,173





#3
Jan2713, 11:04 AM

P: 245

But it's still confusing. What if n=4 (for example)? Then [tex]F_{X_n} = 1[/tex] if [tex]x \geq 1/4[/tex], and [tex]F_{X_n}=0[/tex], when [tex]x < 1/4[/tex], right? So for any x between 0 and 1/4, the limit at those points is 0, but the limit of [tex]F_X[/tex] at those points is 1...so the limits are not equal, are they? 



#4
Jan2713, 07:23 PM

Sci Advisor
P: 3,173

Cdf of a discrete random variable and convergence of distributions... 


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