help with random variable convergence

#1
P(Yn=k)= ( (1-2^(-n-1))^(-1) )/ 2^(k+1) for k=0,1,2,...,n. The random variable Y has the geometric distribution where lambda is 1/2. How can I prove that {Yn} converges in distribution to Y?


Any help is appriciated.


I get fY(y)=1/2^(y+1)
but how do I prove the convergence?
 

BGM

TS Contributor
#2
\( \Pr\{Y_n = k\} = \frac {1} {1 - 2^{-n-1}} \frac {1} {2^{k+1}},
k = 0, 1, 2, ..., n \)

Then the cumulative distribution of \( Y_n \),

\( F_{Y_n}(y) = \Pr\{Y_n \leq y\} = \left\{\begin{array}{lll}
0 & \mbox{if} & y < 0 \\
{\displaystyle \frac {1} {1 - 2^{-n-1}} \left(1 -
\frac {1} {2^{\lfloor y\rfloor + 1} } \right) }& \mbox{if} & 0 \leq y < n \\
1 & \mbox{if} & y \geq n
\end{array}\right . \)

\( \lim_{n\to +\infty} F_{Y_n}(y) = \left\{\begin{array}{lll}
0 & \mbox{if} & y < 0 \\
{\displaystyle 1 - \frac {1} {2^{\lfloor y\rfloor + 1} } }& \mbox{if}
& y \geq 0
\end{array}\right .
= F_Y(y) \)
 
#3
i'm a bit confused here, in your Fyn(y), for the equation where 0<=y<n, why did you put x(1-y) and not 1-(xy)? (x and y represents the parts of Yn)

Also in your Fyn(y), what about 1 if y>=1 since its a cdf ? if you don't add that part, doesn't it make it 1 if y>=0 ?
 
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