Suppose that a scalar random variable y is of the form $y=z+v$, where the pdf of $v$ is $p_{v}(t)=\frac{t}{2}$ on the interval $[0,2]$, and the pdf of $z$ is $p_{z}(t)=2t$on the interval $[0,1]$. Both the densities arezero elsewhere. There is available a single measurement value y=2.5. How to compute the maximum a posteriori estimate of y? I obtained the $pdf$ of $y$ is equal $\frac{y^3}{6}$ on interval $[0,2]$ and zero elsewhere.
Can anyone help me to compute this?Thanks.
You might have some mistake in your problem set-up. As it stands the PDF of y is fully known - typically estimation is used to estimate unknown parameters. The MAP estimate for a variable with a known distribution I would take to be just the value of that variable that maximizes its PDF.
The fact that you are given a measurement for y is irrelevant since you know the distribution of y itself - knowledge of a sample from the fixed distribution does not change the distribution.
Assuming there is not a mistake in this setup, then apparently you need to find the maximum likelihood value of y, given the distribution of y, which is simply the value of y that maximizes its PDF.
The MAP estimate is given: $$ argmax_yP(\text{Data}|y)P(y) \\ =argmax_y\text{Constant}P(y) $$ since the previous data sample is independent of the next data sample from that distribution.
To do this find the maximum of your the PDF of y - I would suggest plotting it first.