# Convergence in Probability

A. If g is a real-valued function that is continuous at a, then g(a^) converges in probability to g(a).

b. Let Y1, Y2, ...Yn be independent random variables, each with probability density function for 0 < y <1, 0 elsewhere. Show that converges in probability to some constant and find the constant.

https://brainmass.com/statistics/probability/convergence-probability-mathematical-statistics-192937

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Please see the attachment for solution.

1. Proofs

a. Prove Theorem 9.2(d): If g is a real-valued function that is continuous at , then converges in probability to .

b. Let Y1, Y2, ...Yn be independent random variables, each with probability density function for 0 y 1, 0 elsewhere. Show that converges in probability to some constant and find the constant.

Solution

a) Since is consistent for , given and , however small, we can find an such that

, whenever . (1)

Again, since g is a real-valued function that is continuous at , given , we can choose such that

Thus

[From (1)]

That is, , whenever .

Hence converges in probability to .

[Note that means and hence or equivalently ]

b) Since the p.d.f. of Y is given by

for 0 y 1, 0 elsewhere

we have

=

=

and hence =

Let .

Then

and

Since and as , converges in probability to the constant = 0.75

[Note that if is a sequence of estimators of such that and as , then is a consistent estimator of ]

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