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I have an exercise, where I am given 5 data points:

x1 = 10, x2 = 7, x3 = 1, x4 = 15, x5 = 8 generated independently.

For the first part, I am told that they follow a Poisson distribution, of parameter theta and I am asked to find the maximum likelihood estimate for theta.

I calculated argmax(theta) of ln(x1,x2,x3,x4,x5 | theta) and got a result of

theta = 41/5 = 8.2.

For the second part I am asked the same thing, but this time I am told that they follow an Exponential distribution of paramter theta.

I did the same calculus and got a result of

theta = 5/41 = 0.12.

Now I am asked which of these two distributions(Poisson or Exponential) is the most probable to have generated the 5 points (x1,x2,x3,x4,x5).

Basically I need to find out which of these two distributions has the highest probability to have generated the 5 points, based (I believe) on that theta that I calculated for both.

But I can't seem to figure it out what's the form of these 2 probabilities that I need to find. Is it the MAP probability? P(theta | x1,x2,x3,x4,x5) ? If yes, I can use the Bayes formula to get

P(x1,x2,x3,x4,x5 | theta) * P(theta) / P(x1,x2,x3,x4,x5). But what is P(theta) and P(x1,x2,x3,x4,x5) ?

Any ideas?

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    I think you can compare the MLE probability ($p(x_1, x_2, x_3, x_4, x_5; \theta)$) between the two and conclude that it is more probable the distribution with the higher MLE probability generated your 5 points. – 2017-01-23
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    So basically I need to calculate P(x1,x2,x3,x4,x5 | theta) for the Poisson distribution with the theta that I calculated (8.2) and then P(x1,x2,x3,x4,x5 | theta) for the Exponential distribution with the other theta(0.12) and choose the distribution with the highest probability? – 2017-01-23
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    Yes, I think so. The MAP estimate is useful when you want to incorporate some prior knowledge about the parameter you're estimating (i.e. $p(\theta)$) into your estimate, but I don't see any prior knowledge in the problem statement. – 2017-01-23
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    If the data points are all integers (rather than rounded to integers) then that makes a discrete Poisson distribution more likely than a continuous Exponential distribution. On the other hand, a Geometric distribution ... – 2017-01-23
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    So are you saying that the Poisson distribution is the most likely to have generated the points? Because I did the math and for the Poisson distribution I got something like exp^-40 * Product(8^xi / xi!) for theta = 8 and every x. And this is clearly very very close to 0. And for the eponential I get something like Product(0.1 * exp^(-0.1x)) for every x which is something like 0.006. So comparing the values, the exponential distribution has a higher probability. But that's only if I did the math right! :D – 2017-01-23
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    For the Poisson distribution I used: Product(e^-8 * 8^xi / xi!) = e^-40 * Product( 8^xi / xi!) =~ 0. – 2017-01-23
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    And for the Exponential I used Product(0.1 * e^0.1xi) = 0.1^5 * Product(e^0.1xi) = 0.0001 * e^1 * e^0.7 * e^0.1 * e^1.5 * e^0.8 = 0.0001 * 2.71 * 2.01 * 1.1 * 4.48 * 2.22 =~ 0.006. – 2017-01-23

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