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Nonnegative random variable X has absolute continuous distribution on interval $[0,a]$. Find

(a) Impartial and consistent estimation $\hat p_1$ of probability $P[0 \le X \le 1.]$. Show impartiality and consistency of $\hat p_1$

what I know is:

We have a absolute continuous distribution with defined interval. It seems to me as Uniform distribution. Then the probability density function could looks like this:

$f(x)=\frac{1}{a}, \quad x \in [0,a]$

$f(x)=0 \qquad$ else

and $F(X)=\frac{x}{a}$

Then I will compute the estimate from this $P[0 \le \frac{x}{a} \le 1]$ right?

Definition (Point estimation):

Let $X_1, X_2, ... , X_n$ be random sample with distribution $P_\theta$ from parametric class $\mathcal P$ and with unknown parameter $\theta$. Point estimation $\theta$ is function $T: \mathbb R^n \rightarrow Θ, T(X_1, ..., X_n) \rightarrow ...,$ description doesn't depend on parameter $\theta$

  1. Point estimation $T$ is impartial, if $\forall \theta \in Θ$ is $ET = \theta$

  2. Point estimation $T$ is consistent, if $\forall \theta \in Θ$ is $P[T- \theta] \gt \epsilon \rightarrow 0 $

How can I show these two attributes?

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  • $\begingroup$ I don't understand the question. What are you trying to find a point estimate of? What does "estimation $\hat p_1$ with probability $P[0 \le X \le 1]$" mean? $\endgroup$ – Blaza Feb 1 '17 at 14:02
  • $\begingroup$ I apologise. The formulation should be: Find the point estimation p of probability P[0≤X≤1] $\endgroup$ – Muffy Feb 1 '17 at 14:13

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