MATH 262 Probability Theory
16 March 2026

Computing the skewness coefficient from a moment generating function:

Skewness coefficient is γ=((E[(((X-μ))^3)])/(σ^3)).

Expand[(((x-μ))^3)]

So E[(((X-μ))^3)]=E[(X^3)]-3E[(X^2)]E[X]+3E[X]((E[X])^2)-((E[X])^3)=E[(X^3)]-3E[(X^2)]E[X]+2((E[X])^3).

Also (σ^3)=(((E[(X^2)]-((E[X])^2)))^(3/2)). Thus, we can compute the skewness coefficient in terms of the moments of X.

Binomial Distribution

binomial random variable with n = 10 and p = (1/2):

m[t_]:=(1-1/2+1/2Exp[t])^10

Here is the first moment:

m'[0]

Now compute the skewness coefficient:

(m'''[0]-3m''[0]m'[0]+2(m'[0])^3)/(m''[0]-(m'[0])^2)^(3/2)

This makes sense, because the binomial distribution with p = (1/2) is symmetric.

Geometric Distribution

the mgf of a geometric distribution with p=(1/3):

my[t_]:=Exp[t]/(3-2Exp[t])

my[t]

the moments:

my'[0]

my''[0]

my'''[0]

the skewness coefficient:

(my'''[0]-3my''[0]*my'[0]+2(my'[0])^3)/(my''[0]-my'[0]^2)^(3/2)

N[(5/(Sqrt[6]))]

This distribution is skewed right. The skewness of a Geometric(p) distribution is ((2-p)/(Sqrt[1-p])).

(5/3)/Sqrt[2/3]

Poisson Distribution

mgf of a Poisson distribution with mean 4

mp[t_]:=Exp[4(Exp[t]-1)]

the moments:

mp[t]

mp'[0]

mp''[0]

mp'''[0]

the skewness coefficient:

(mp'''[0]-3mp''[0]*mp'[0]+2(mp'[0])^3)/(mp''[0]-mp'[0]^2)^(3/2)

This distribution is skewed right. Skewness of a Poisson(λ) distribution is (λ^(-1/2)).