# Let $X$ be a single observation from the density $f(x) = (2θx + 1 − θ)I[0,1](x)$ with $−1≤ θ ≤ 1$. Find the most powerful test of size $α$ and its power

• Find a most powerful test of size $α$ of $H_0 : θ = 0$ versus $H_1 : θ = 1$.
• Find a uniformly most powerful test of size $α$ of $H_0 : θ = 0$ versus $H_1 : θ > 0$
• To test $H_0: θ ≤ 0$ versus $H_1: θ > 0$, the following procedure was used: Reject $H_0$ if $X$ exceeds $1/2$. Find the power and size of this test.
• I feel the bounty is a little to low for this.

• I second that.

• Bounty has been increased

• Is this for only one observation (n=1) or for a general random sample of n observations?

• @Rage, this is a good question, yes the original question states let X be a single observation from the density. I will update the question to make this clear.

• Without that hypothesis the problem was too difficult!

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• Hi, just checking on page 2, the derivative (h'(x)) should the second term be -2θb(2θax+1-2θa) instead of -2θ b(2θax+1-2θb)?

• Yes, there was a typo in the solution. The final result does not change, though. Let me add a new version.