PDF, CDF, Mean, Variance, and Moment Generating Function

Discrete Distribution

In[7]:= PDF@BinomialDistribution@n, pD, kD �� TraditionalForm

Out[7]//TraditionalForm=

pk K nk

O H1 - pLn-k 0 £ k £ n

0 True

In[8]:= CDF@BinomialDistribution@n, pD, kD �� TraditionalForm

Out[8]//TraditionalForm=

I1-pHn - dkt, dkt + 1L 0 £ k £ n

1 k > n

In[9]:= Mean@BinomialDistribution@n, pDD

Out[9]= n p

In[10]:= Variance@BinomialDistribution@n, pDD

Out[10]= n H1 - pL p

In[11]:= MomentGeneratingFunction@BinomialDistribution@n, pD, kD

Out[11]= I1 + I-1 + ãkM pMn

In[12]:= D@%, kD

Out[12]= ãk n p I1 + I-1 + ã

kM pM-1+n

In[13]:= FullSimplify@%, k � 0D

Out[13]= n p

In[14]:= PDF@BernoulliDistribution@pD, kD

Out[14]=

1 - p k � 0

p k � 1

0 True

In[15]:= CDF@BernoulliDistribution@pD, kD

Out[15]=

0 k < 0

1 - p 0 £ k < 1

1 True

In[16]:= Mean@BernoulliDistribution@pDD

Out[16]= p

In[17]:= Median@BernoulliDistribution@pDD

Out[17]=1 p >

1

2

0 True

In[18]:= Variance@BernoulliDistribution@pDD

Out[18]= H1 - pL p

In[19]:= MomentGeneratingFunction@BernoulliDistribution@pD, kD

Out[19]= 1 - p + ãk p

In[20]:= D@%, kD

Out[20]= ãk p

In[21]:= FullSimplify@%, k � 0D

Out[21]= p

In[22]:= PDF@HypergeometricDistribution@n, r, ND, kD �� TraditionalForm

Out[22]//TraditionalForm=

r

k

N-r

n-k

K N

nO

0 £ k £ n ì n - N + r £ k £ n ì 0 £ k £ r ì n - N + r £ k £ r

0 True

In[26]:= CDF@HypergeometricDistribution@n, r, ND, kD �� TraditionalForm

Out[26]//TraditionalForm=

1 - Ir ! HN - rL ! 3F�

2H1, -n + dkt + 1, -r + dkt + 1; dkt + 2, -n + N - r + dkt + 2; 1LM �KK N

nO H-dkt + n - 1L ! H-dkt + r - 1L !O

0 £ k ì n - N + r £ k ì k < n ì k < r

1 k ³ n ê k ³ r

In[27]:= Mean@HypergeometricDistribution@n, r, NDD

Out[27]=n r

N

In[28]:= Variance@HypergeometricDistribution@n, r, NDD

Out[28]=

n H-n + NL r I1 -r

NM

H-1 + NL N

In[29]:= MomentGeneratingFunction@HypergeometricDistribution@n, r, ND, kD

Out[29]= Hypergeometric2F1A-n, -r, -N, 1 - ãkE

In[30]:= D@%, kD

Out[30]=ãk n r Hypergeometric2F1@1 - n, 1 - r, 1 - N, 1 - ãkD

N

In[31]:= FullSimplify@%, k � 0D

Out[31]=n r

N

In[32]:= PDF@GeometricDistribution@pD, kD

Out[32]=H1 - pLk p k ³ 0

0 True

In[33]:= CDF@GeometricDistribution@pD, kD �� TraditionalForm

Out[33]//TraditionalForm=

1 - H1 - pLdkt+1 k ³ 0

0 True

In[34]:= Mean@GeometricDistribution@pDD

Out[34]= -1 +

1

p

In[35]:= Variance@GeometricDistribution@pDD

Out[35]=1 - p

p2

2 PDF, CDF, Mean, Variance , & Moment Generating Function.nb

In[36]:= MomentGeneratingFunction@GeometricDistribution@pD, kD

Out[36]=p

1 - ãk H1 - pLIn[37]:= D@%, kD

Out[37]=ãk H1 - pL p

H1 - ãk H1 - pLL2

In[38]:= FullSimplify@%, k � 0D

Out[38]= -1 +

1

p

In[39]:= PDF@PoissonDistribution@ΜD, kD

Out[39]=

ã-Μ Μk

k!k ³ 0

0 True

In[40]:= CDF@PoissonDistribution@ΜD, kD �� TraditionalForm

Out[40]//TraditionalForm=

¶ QHdkt + 1, ΜL k ³ 00 True

In[41]:= Mean@PoissonDistribution@ΜDD

Out[41]= Μ

In[42]:= Variance@PoissonDistribution@ΜDD

Out[42]= Μ

In[43]:= MomentGeneratingFunction@PoissonDistribution@ΜD, kD

Out[43]= ãI-1+ãkM Μ

In[44]:= D@%, kD

Out[44]= ãk+I-1+ãkM Μ

Μ

In[45]:= FullSimplify@%, k � 0D

Out[45]= Μ

In[46]:= PDF@NegativeBinomialDistribution@n, pD, kD �� TraditionalForm

Out[46]//TraditionalForm=

H1 - pLk pn K k + n - 1n - 1

O k ³ 0

0 True

In[47]:= CDF@NegativeBinomialDistribution@n, pD, kD �� TraditionalForm

Out[47]//TraditionalForm=

IpHn, dkt + 1L k ³ 0

0 True

In[48]:= Mean@NegativeBinomialDistribution@n, pDD

Out[48]=n H1 - pL

p

PDF, CDF, Mean, Variance , & Moment Generating Function.nb 3

In[49]:= Variance@NegativeBinomialDistribution@n, pDD

Out[49]=n H1 - pL

p2

In[50]:= MomentGeneratingFunction@NegativeBinomialDistribution@n, pD, kD

Out[50]=p

1 - ãk H1 - pL

n

In[51]:= D@%, kD

Out[51]=

ãk n H1 - pL p J p

1-ãk H1-pL N-1+n

H1 - ãk H1 - pLL2

In[52]:= FullSimplify@%, k � 0D

Out[52]= n -1 +

1

p

Continuous Distribution

In[53]:= PDF@NormalDistribution@Μ, ΣD, kD

Out[53]=ã

-Hk-ΜL2

2 Σ2

2 Π Σ

In[54]:= CDF@NormalDistribution@Μ, ΣD, kD �� TraditionalForm

Out[54]//TraditionalForm=

1

2erfc

Μ - k

2 Σ

In[55]:= Mean@NormalDistribution@Μ, ΣDD

Out[55]= Μ

In[56]:= Median@NormalDistribution@Μ, ΣDD

Out[56]= Μ

In[57]:= Variance@NormalDistribution@Μ, ΣDD

Out[57]= Σ2

In[58]:= MomentGeneratingFunction@NormalDistribution@Μ, ΣD, kD

Out[58]= ãk Μ+

k2 Σ2

2

In[59]:= D@%, kD

Out[59]= ãk Μ+

k2 Σ2

2 IΜ + k Σ2M

In[60]:= FullSimplify@%, k � 0D

Out[60]= Μ

4 PDF, CDF, Mean, Variance , & Moment Generating Function.nb

In[61]:= PDF@UniformDistribution@8a, b<D, kD

Out[61]=

1

-a+ba £ k £ b

0 True

In[62]:= CDF@UniformDistribution@8a, b<D, kD

Out[62]=

-a+k

-a+ba £ k £ b

1 k > b

0 True

In[63]:= Mean@UniformDistribution@8a, b<DD

Out[63]=a + b

2

In[64]:= Variance@UniformDistribution@8a, b<DD

Out[64]=1

12H-a + bL2

In[65]:= MomentGeneratingFunction@UniformDistribution@8a, b<D, kD

Out[65]=-ãa k + ãb k

H-a + bL k

In[66]:= D@%, kD

Out[66]= -

-ãa k + ãb k

H-a + bL k2+

-a ãa k + b ãb k

H-a + bL k

In[67]:= FullSimplify@%, k � 0D

FullSimplify::infd : Expression -

-ãa k

+ ãb k

H-a + bL k2

+

-a ãa k

+ b ãb k

H-a + bL k

simplified to Indeterminate. �

Out[67]= Indeterminate

In[68]:= PDF@ExponentialDistribution@ΛD, kD

Out[68]=ã-k Λ Λ k ³ 0

0 True

In[69]:= CDF@ExponentialDistribution@ΛD, kD

Out[69]=1 - ã-k Λ k ³ 0

0 True

In[70]:= Mean@ExponentialDistribution@ΛDD

Out[70]=1

Λ

In[71]:= Median@ExponentialDistribution@ΛDD

Out[71]=Log@2D

Λ

In[72]:= Variance@ExponentialDistribution@ΛDD

Out[72]=1

Λ2

PDF, CDF, Mean, Variance , & Moment Generating Function.nb 5

In[73]:= MomentGeneratingFunction@ExponentialDistribution@ΛD, kD

Out[73]=Λ

-k + Λ

In[74]:= D@%, kD

Out[74]=Λ

H-k + ΛL2

In[75]:= FullSimplify@%, k � 0D

Out[75]=1

Λ

In[76]:= PDF@GammaDistribution@Α, ΒD, kD

Out[76]=ã

-k

Β k-1+Α Β-Α

Gamma@ΑD k > 0

0 True

In[77]:= CDF@GammaDistribution@Α, ΒD, kD �� TraditionalForm

Out[77]//TraditionalForm=

QJΑ, 0,k

ΒN k > 0

0 True

In[78]:= Mean@GammaDistribution@Α, ΒDD

Out[78]= Α Β

In[79]:= Median@GammaDistribution@Α, ΒDD �� TraditionalForm

Out[79]//TraditionalForm=

Β Q-1Α, 0,

1

2

In[80]:= Variance@GammaDistribution@Α, ΒDD

Out[80]= Α Β2

In[81]:= MomentGeneratingFunction@GammaDistribution@Α, ΒD, kD

Out[81]= H1 - k ΒL-Α

In[82]:= D@%, kD

Out[82]= Α Β H1 - k ΒL-1-Α

In[83]:= FullSimplify@%, k � 0D

Out[83]= Α Β

In[101]:= PDF@RayleighDistribution@aD, kD

Out[101]=ã

-k2

2 a2 k

a2k > 0

0 True

In[102]:= CDF@RayleighDistribution@aD, kD

Out[102]= 1 - ã-

k2

2 a2 k > 00 True

6 PDF, CDF, Mean, Variance , & Moment Generating Function.nb

In[103]:= Mean@RayleighDistribution@aDD

Out[103]= aΠ

2

In[106]:= Median@RayleighDistribution@aDD

Out[106]= a Log@4D

In[104]:= Variance@RayleighDistribution@aDD

Out[104]= a2 2 -

Π

2

In[113]:= MomentGeneratingFunction@RayleighDistribution@aD, kD �� TraditionalForm

Out[113]//TraditionalForm=

Π

2a k ã

a2 k2

2 erfa k

2+ 1 + 1

In[114]:= D@%, kD

Out[114]= a2 k + a ã

a2 k2

2

Π

21 + ErfB

a k

2F + a3

ã

a2 k2

2 k2Π

21 + ErfB

a k

2F

In[115]:= Simplify@%, k � 0D

Out[115]= aΠ

2

In[84]:= PDF@ChiSquareDistribution@ΝD, kD �� TraditionalForm

Out[84]//TraditionalForm=

ã-k�2 2-Ν�2 kΝ

2-1

GI Ν

2M k > 0

0 True

In[85]:= CDF@ChiSquareDistribution@ΝD, kD

Out[85]=GammaRegularizedA Ν

2, 0,

k

2E k > 0

0 True

In[86]:= Mean@ChiSquareDistribution@ΝDD

Out[86]= Ν

In[87]:= Variance@ChiSquareDistribution@ΝDD

Out[87]= 2 Ν

In[88]:= MomentGeneratingFunction@ChiSquareDistribution@ΝD, kD

Out[88]= H1 - 2 kL-Ν�2

In[89]:= D@%, kD

Out[89]= H1 - 2 kL-1-Ν

2 Ν

In[90]:= FullSimplify@%, k � 0D

Out[90]= Ν

PDF, CDF, Mean, Variance , & Moment Generating Function.nb 7

In[91]:= PDF@LogNormalDistribution@Μ, ΣD, kD �� TraditionalForm

Out[91]//TraditionalForm=

ã

-HlogHkL-ΜL2

2 Σ2

2 Π k Σ

k > 0

0 True

In[92]:= CDF@LogNormalDistribution@Μ, ΣD, kD �� TraditionalForm

Out[92]//TraditionalForm=

1

2erfcK Μ-logHkL

2 Σ

O k > 0

0 True

In[93]:= Mean@LogNormalDistribution@Μ, ΣDD

Out[93]= ãΜ+

Σ2

2

In[94]:= Median@LogNormalDistribution@Μ, ΣDD

Out[94]= ãΜ

In[95]:= Variance@LogNormalDistribution@Μ, ΣDD

Out[95]= ã2 Μ+Σ2 I-1 + ã

Σ2M

In[96]:= PDF@WeibullDistribution@Α, ΒD, kD

Out[96]=

ã-

k

Β

Α

Α J k

ΒN

-1+Α

Βk > 0

0 True

In[97]:= CDF@WeibullDistribution@Α, ΒD, kD

Out[97]= 1 - ã-J k

ΒN

Α

k > 00 True

In[98]:= Mean@WeibullDistribution@Α, ΒDD

Out[98]= Β GammaB1 +

1

Α

F

In[99]:= Variance@WeibullDistribution@Α, ΒDD

Out[99]= Β2

-GammaB1 +

1

Α

F2

+ GammaB1 +

2

Α

F

8 PDF, CDF, Mean, Variance , & Moment Generating Function.nb