Poisson Distribution - Radioactivity

https://isaacphysics.org/questions/poisson_radioactivity?board=9d46f10a-d2cd-446b-a584-ca16b5f57212&stage=further_a

For part c, what is the difference between 'expected value' and 'most probable value' and why aren't they the same thing?

I initially thought it would just be the peak of the graph so I just plotted it on desmos and used the peak x-value but that didnt seem to work?

Any help always appreciated.

Reply 1

mqb2766

Original post by mosaurlodon

https://isaacphysics.org/questions/poisson_radioactivity?board=9d46f10a-d2cd-446b-a584-ca16b5f57212&stage=further_a
For part c, what is the difference between 'expected value' and 'most probable value' and why aren't they the same thing?
I initially thought it would just be the peak of the graph so I just plotted it on desmos and used the peak x-value but that didnt seem to work?
Any help always appreciated.

Theyre the usual definitions, though b) is for rays arriving and c) is for rays detected which are different random variables due to the saturation. The expected value is a standard a level concept and for a discrete random variable is the sum over n of
x_n p(x_n)

The most probable (mode, integer) and the expected value (mean, real) for a standard poisson are related though
https://en.wikipedia.org/wiki/Poisson_distribution
You can work it out without calculating the actual probabilities (hint 3, b).

The famous example of the mode and mean being different is that the mean number of legs a person has is something like 1.995 as noone has more than 2 but a few people have 0 or 1. The mode is obviously 2.

(edited 1 year ago)

Reply 2

mosaurlodon

Oh so the expected value is just the value for lambda?

edit: sorry I just read your edit give me a minute

(edited 1 year ago)

Reply 3

mqb2766

Original post by mosaurlodon

Oh so the expected value is just the value for lambda?

edit: sorry I just read your edit give me a minute

For a standard poisson distribution the mean or expected value is lambda. However, for c) its a slighlty modified poisson distribution due to the saturation and the mean will be a bit less than lambda.

Reply 4

mosaurlodon

Wait actually im a little confused about the standard expected value part - using desmos I just did P(x_n)*x_n and summed it but this seems to approach 0 rather than lambda.

I also dont really get how the saturation part works - I know intuitively it shifts the entire graph to the left, since the lower values must be more probable but dont really know by what factor it shifts?

Reply 5

mqb2766

90*0.03777 = 3.4
When you calculcate the expected value, you dont divide by "t", its the random value multiplied by the probabilitiy (summed).

For the saturation, the new p(5) will be the old P(X>=5) for the poisson as all values of 5 or greater are mapped to 5. Thats why part a) gets you to calculate the probability of saturation.