How Does the Constraint θ ≤ 1/4 Affect the MLE in Bernoulli Trials?

[SandMan249]

Homework Statement

Two independent bernoulli trials resulted in one failure and one success. What is the MLE of the probability of success θ is it is know that θ is at most 1/4

Homework Equations

f(x,θ) = θ^x (1-θ)^1-x

The Attempt at a Solution

Now, I know how to find the likelihood and use it to solve for the MLE. But I am not sure how the "θ is at most 1/4" would factor into the equation.

For a Bernoulli trial: f(x,θ) = θ^x (1-θ)^1-x
L(θ) = θ(1-θ)
L'(θ) = 1-2θ ----> equate to zero
θ(hat) = 1/2 (which is the MLE)

But what do I do about the fact that "θ is at most 1/4"? Please help

 

[Office_Shredder]

You're trying to maximize L(θ) on the interval [0,1/4]. Recall a basic calculus fact that the maximum of a function on a closed interval is either a critical point (where the derivative is zero) or an end point

 

[SandMan249]

Thank you.
In this case, since the function is simply: θ(hat) = 1/2 (a constant)
The maximum will lie at 1/4
Correct?

 

[Office_Shredder]

The critical point is at 1/2. But the function you want to maximize is L(θ) = θ(1-θ)