Use the counter example to show that convergence in distribution does not imply convergence in probability.

Let Ω={1, 2}, Ϝ=ρ(Ω), and let P be defined by P({1})=P({2})=1/2.
Let X_n(1)=1, X_n(2)=0 (for all n) and X(1)=0, X(2)=1.

Then show that X_n converges to X in distribution but not in probability.


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