Random Variables and Distributions Flashcards
Sample vs population - which letters to denote?
sample is a subgroup of population
Greek letters for properties of POPULATION
roman letters for properties of SAMPLE
define probability
predicting properties of a SAMPLE
define statistics
deducing information about the POPULATION from the sample
define cumulative probability (both discrete and continuous)
F(x) = P(X≤x)
sum of all probabilities up to X=x for discrete
integration from -inf to x for continuous
CDF F(x) obtained by integrating PDF f(x), vice versa by differentiating
manipulating expectations:
E[aX+b] =
a E[X] + b
expectation of a sum E[A+B] is equal to…
the sum of expectations
= E[A] + E[B]
Manipulating variances: Var[aX+b] =
a^2 Var[X]
proof by expanding Var[X] = E[X^2] - (E[X])^2 definition
under what conditions is it true that Var[X+Y] = Var[X] + Var[Y] = Var[X-Y] ?
only for INDEPENDENT X and Y
standard deviation quantifies the…
width of a probability distribution
Define skewness. How is it calculated? What does a -ve, +ve and skewness of 0 mean?
skewness is a measure of the asymmetry of a distribution.
Skew[X] = E[[X-µ)^3] / σ^3
negative skew = tail to the left
positive skew = tail to the right
0 skew usually symmetric
define kurtosis. how is it calculated?
kurtosis is a measurement of how much WEIGHT of a distribution lies in its TAILS
(ie. how likely it is to observe extreme values)
Kurt[X] = E[(X-µ)^4] / σ^4
define excess kurtosis - how is it calculated?
comparing kurtosis to that of a normal distribution, ie. how much more weight is in the tails
XS Kurt = Kurt[X] - 3
-ve XS kurt means less weight is in the tails compared to normal
define moment of a distribution. how is the nth moment calculated?
nth moment of a distribution:
M(n) = E[X^n]
what is a moment generating function?
some function m(t) such that the limit as t–>0 of the nth derivative wrt t gives the nth moment, M(n).
how to find the moment generating function m(t)
m(t) = E[exp(tX)] = integral of exp(tx) f(x) for some continuous PDF f(x)
what does the random variable for the Binomial distribution describe?
the number of times, m, that a particular event happens in n independent measurements
constant probability of success, p
only two outcomes
the random variable in the Poisson distribution describes…
the number of times, m, a random event occurs in a specified time interval.
Constant chance of occurrence.
probability of event occurring is proportional to length of time period.
will often be given the avg number of occurrences per interval (ie. lambda)
examples of exponential distributions
residence time of molecules in a CSTR
lifetime of reactant molecules in batch reactor for first order reaction
moment generating function of exponential distributions
m(t) = λ/λ-t
how to change variables for normal distribution to standardise to 𝛷(z)
z = (x-µ) / σ
new random variable: Z = (X-µ)/σ
such that Z~N(0,1) which has a CDF of 𝛷(z)
where X~N(µ,σ)
a binomial dist can be approximated as normal if…
np > 5 and
nq = n(1-p) >5
a poisson dist can be approximated as normal if…
lamda > 15
when approximating a discrete distribution as continuous (normal), one must…
APPLY A CONTINUITY CORRECTION
less than –> 0.5 higher
more than –> 0.5 lower
Bayes’ Theorem
think probability tree branches
P(A|B) = P(B|A) * P(A)/P(B)
