I have encountered a problem that seemed a bit odd to me:
Let Yn = 2Xn where Xn is distributed Binomially B(p ,10), p < 1/20 prove or disprove that Yn converges in probability to 20p.
I tried playing around with chevishev, markov, and hoeffding inequality but all of those only give the lower bound, so i think they cannot imply convergence. On the other hand, I have not figured a way how to disprove convergence in probability. any ideas ?
Note that the convergence in probability would imply that $Y_n$ converges in distribution to $20p$. However, note that $Y_n\stackrel{d}{\to} 2Z$ where $Z$ is a random variable such that $Z\sim \text{Bin}(p, 10)$. We have reached a contradiction and it follows that $Y_n$ does not converge in prob to $20p$.