As on the previous page, let X1,…,Xn be i.i.d. with pdf

fθ(x)=θxθ−11(0≤x≤1)
where θ>0.

(a)
2 puntos posibles (calificables, resultados ocultos)
Assume we do not actually get to observe X1,…,Xn. Instead let Y1,…,Yn be our observations where Yi=1(Xi≤0.5). Our goal is to estimate θ based on this new data.

What distribution does Yi follow?

First, choose the type of the distribution:

Bernoulli
Poisson
Normal
Exponential
sin responder
Second, enter the parameter of this distribution in terms of θ. Denote this parameter by mθ. (If the distribution is normal, enter only 1 parameter, the mean).

follows the Bernoulli

the second is 1/(2^theta)

To determine the distribution followed by Yi, we need to understand what Yi represents in the given problem.

Yi takes the value 1 if Xi is less than or equal to 0.5, and it takes the value 0 otherwise. In other words, Yi acts as an indicator or a binary variable that tells us whether a specific observation Xi is less than or equal to 0.5.

Given this definition, it is clear that Yi follows a Bernoulli distribution. In a Bernoulli distribution, the random variable can take one of two possible outcomes (0 or 1) with a certain probability of success (p) for the event of interest. In this case, the event of interest is whether Xi is less than or equal to 0.5.

Therefore, the distribution followed by Yi is Bernoulli.

Now, let's determine the parameter of this Bernoulli distribution in terms of θ. The parameter p in the Bernoulli distribution represents the probability of success. In this case, success means Xi being less than or equal to 0.5.

Since Xi follows a continuous distribution with a probability density function (pdf), we can find the probability of Xi being less than or equal to 0.5 by integrating the pdf from 0 to 0.5. This will give us the probability of success, which is equivalent to the parameter p of the Bernoulli distribution.

Let's perform the integration to find the parameter of the Bernoulli distribution:

∫[0 to 0.5] θx^(θ-1) dx

To integrate this, we can use the power rule of integration:

(θx^θ) / θ | [0 to 0.5]

Using the limits of integration:

(θ*(0.5)^θ - θ*(0)^θ) / θ

Simplifying:

(0.5)^θ - 0^θ

As anything raised to the power of 0 is 1, we get:

(0.5)^θ - 1

Therefore, the parameter of the Bernoulli distribution in terms of θ is mθ = (0.5)^θ - 1.

So, the answer is:
- The distribution followed by Yi is Bernoulli.
- The parameter of this distribution in terms of θ is mθ = (0.5)^θ - 1.