Great video. Thank you! There is something that confuses me. Shouldn’t the standard deviation and the mean value have the same dimension? If the variable is some physical magnitud, for example a surface in square meters, the standard deviation would turn out to be in meters. This doesn’t make much sense. What am I missing?
@ThinkLikeaPhysicist
Жыл бұрын
Hi! Well, the Poisson distribution itself gives you the probability of seeing n events in an interval when, on average, lambda are expected. But, n is just a number of events; it doesn't have units. We could define a related distribution, though. Let's say we have a Poisson distribution that describes, say, how many bottles of a product get broken during a shipment. (In other words, we send shipments of a large number of bottles of some product. On average, lambda get broken in transport. What is the probability that n bottles get broken in a specific shipment?) The number n is what is described by a Poisson distribution; it would have a standard deviation of sqrt(lambda). But, now, let's say each bottle contains 1 kg of stuff. We can write down a probability distribution for the mass of stuff that is lost in a given shipment. This would look just like a Poisson distribution, except that the x-axis is multiplied by 1 kg. For this latter distribution, the mean is lambda*1kg, and the standard deviation is sqrt(lambda)*1kg. It might become slightly clearer if we note that 1kg = 1000g, and we could also write the distribution using g instead of kg. Then, our probability distribution for the amount of stuff in broken bottles looks just like a Poisson distribution, except the x-axis is scaled by 1000g. Its mean is 1000g*lambda, and its standard deviation is 1000g*sqrt(lambda). Note that we don't take the square root of the units, just like we don't take the square root of the number 1000. Does that clear things up?
@enricolucarelli816
Жыл бұрын
@@ThinkLikeaPhysicist YES! THANK YOU! 👍 I guess this reasoning is extendable to any random variable, even a continuos one: it represents a count of events, and not a magnitude, thus it has to be considered dimensionless. This also clarifies something I had trouble with since a long time: the issue with the entropy of a continuos random variable. It is the integral of a probability density times the logarithm of the same probability density. The dimensional analysis with something inside a logarithm just didn’t make any sense to me. Now I think I understand.
@rishabhgupta7264
10 ай бұрын
But why did we take HWHM and not FWHM for determination of sigma.
@ThinkLikeaPhysicist
10 ай бұрын
Hi! Actually, we did neither. The HWHM and FWHM are comparatively easy to figure out by looking at a distribution, but sigma is the standard deviation, which is actually yet another measure of the width of the distribution. It's not so easy to read it off a plot, but it's mathematically useful, especially if you're looking at the very common gaussian or normal distribution.
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