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:explode:DISTRIBUTIONS:explode: (Discrete random variable (Probability…
:explode:DISTRIBUTIONS:explode:
Discrete random variable
we are gonna look at the distribution of a particular random variable.
Probability Distribution
used to describe possible vals of a random var X
Distributions can be summarised by mean, variance, moments, spread
Probability Mass Function
a function that gives the probability that a discrete random variable is exactly equal to some value.
Cumulative Distribution Function
is the probability that X will take a value less than or equal to x.
Terminology
Mean, Expected Value
μ=E[X] = SUM ( X *p(x))
Alternatively, if expected value was a function h(x), then E[X} = SUM (h(x)*p(x))
Moment
Tells about shape of a set of points.
Varience
Independent and Identical Bernoulli Trials
a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.
A random variable corresponding to a binomial is denoted by B ( n , p ), and is said to have a
binomial distribution.
Continuous Random Variable
Distributions
Cumulative Distribution Function
IF f(x) = 1 / (b-a)
THEN it is a
uniform random variable
So MEAN = E(X) = (a+b) / 2
VAR = V(X) = (b-a)^2 / 12
Probability Density Function
IF it has a particular complicated looking PDF, THEN it is a
normal random variable
If MEAN = 0, VAR = 1 then it is a
standard normal variable
called Z
Exponential Distribution
is given by a survival function involving LAMBDA
describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.
exhibits
lack of memory
property
distribution of a "waiting time" until a certain event, does not depend on how much time has elapsed already.
eg, radioactive decay of particles
For accurate modeling, we must constantly 'forget' which state the system is in: the probabilities would not be influenced by the history of the process