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PMF f(x)=P(X=x)
gives the probability at each discrete point. CDF F(x)=P(X≤x)
accumulates probabilities up to x. For discrete X, F(x)
jumps at the support points and is flat between them.
Two things: (1) all f(x) ≥ 0
, and (2) the total sums to 1. In Excel, if x are in A2:A10 and probabilities in B2:B10, check =SUM(B2:B10)
equals 1 (within rounding).
Mean: E[X] = Σ x f(x)
. Variance: Var(X) = E[X^2] − (E[X])^2
where E[X^2] = Σ x^2 f(x)
. In Excel: =SUMPRODUCT(A2:A10,B2:B10)
for E[X], and =SUMPRODUCT((A2:A10)^2,B2:B10) - (SUMPRODUCT(A2:A10,B2:B10))^2
for Var(X).
E[Y] = aE[X] + b
and Var(Y) = a^2 Var(X)
. Shifts (b
) don’t change spread; scaling by a
stretches the spread by |a|
(variance scales by a^2
).
Use Binomial(n,p) when you count successes in n independent, identical trials with success probability p. Mean: np
, Variance: np(1−p)
. Example: correct T/F answers when guessing.
Number of trials to the first success (support 1,2,3,…). PMF: P(X=k)=(1−p)^{k−1}p
. Mean: 1/p
. It’s memoryless: P(X>s+t | X>s)=P(X>t)
.
Negative Binomial counts trials to achieve r successes (geometric is the case r=1). PMF: P(X=k)=C(k−1,r−1)p^r(1−p)^{k−r}
, mean: r/p
, variance: r(1−p)/p^2
. Binomial counts number of successes in a fixed number of trials.
Counts of events in a fixed interval with constant average rate λ
and independent occurrences. PMF: e^{−λ}λ^x/x!
, mean = variance = λ
. Also a good approx. to Binomial when n
is large and p
is small with λ=np
.
If X
takes integers from a
to b
equally likely, then f(x)=1/(b−a+1)
, E[X]=(a+b)/2
, and Var(X)={ (b−a+1)^2 − 1 }/12
.
np ≥ 10
and n(1−p) ≥ 10
(use continuity correction).n
large, p
small, with λ=np
(moderate λ).λ
with mean = variance = λ
.