1. Definition and classification of random variables
Definition: A random variable is a variable that takes on different values in a random experiment.
Categorical random variable (Nominal)
Examples: Gender (male, female), occupation (civil servant, corporate employee, student, retired, unemployed), test result (negative, positive)
Ordered categorical random variables
Examples: attitude (strongly agree, agree, neutral, disagree, strongly disagree), frequency of use (once a week, once every two weeks, once every six months, rarely, never)
Numeric random variables
Example: Age (13, 14, 15, 16, etc.), Income (any value can be filled in)
2. Discrete Probability Distribution
2.1 Binomial Distribution
definition: Description in
n
n
n Success in independent trials
k
k
k The probability of success in each trial is
p
p
p。
formula：
X
∼
Bin
(
n
,
p
)
X sim text{Bin}(n, p)
X∼Bin(n,p)
Probability Mass Function (PMF)：
P
(
X
=
k
)
=
(
n
k
)
p
k
(
1
−
p
)
n
−
k
P(X = k) = binom{n}{k} p^k (1-p)^{n-k}
P(X=k)=(kn)pk(1−p)n−k
Cumulative Distribution Function (CDF)：
F
(
X
=
k
)
=
∑
i
=
0
k
(
n
i
)
p
i
(
1
−
p
)
n
−
i
F(X = k) = sum_{i=0}^{k} binom{n}{i} p^i (1-p)^{n-i}
F(X=k)=∑i=0k(in)pi(1−p)n−i
2.2 Bernoulli Distribution
definition: describes the probability of success (or failure) in a trial. The probability of success is
p
p
p。
formula：
X
∼
Bern
(
p
)
X sim text{Bern}(p)
X∼Bern(p)
Probability Mass Function (PMF)：
P
(
X
=
x
)
=
{
p
if
x
=
1
1
−
p
if
x
=
0
P(X = x) = {pif x=11−pif x=0P(X=x)={p1−pif x=1if x=0
Cumulative Distribution Function (CDF)：
F
(
X
=
x
)
=
{
0
if
x
<
0
1
−
p
if
0
≤
x
<
1
1
if
x
≥
1
F(X = x) = {0if x<01−pif 0≤x<11if x≥1F(X=x)=⎩⎨⎧01−p1if x<0if 0≤x<1if x≥1
2.3 Geometric Distribution
definition: describes the probability of the number of failures before the first success, and the probability of success in each trial is
p
p
p。
formula：
X
∼
Geom
(
p
)
X sim text{Geom}(p)
X∼Geom(p)
Probability Mass Function (PMF)：
P
(
X
=
k
)
=
(
1
−
p
)
k
p
P(X = k) = (1-p)^k p
P(X=k)=(1−p)kp
Cumulative Distribution Function (CDF)：
F
(
X
=
k
)
=
1
−
(
1
−
p
)
k
+
1
F(X = k) = 1 - (1-p)^{k+1}
F(X=k)=1−(1−p)k+1
2.4 Negative Binomial Distribution
definition: describes the number of failures before reaching the rth success, and the probability of success in each trial is
p
p
p。
formula：
X
∼
NegBin
(
r
,
p
)
X sim text{NegBin}(r, p)
X∼NegBin(r,p)
Probability Mass Function (PMF)：
P
(
X
=
k
)
=
(
k
+
r
−
1
k
)
p
r
(
1
−
p
)
k
P(X = k) = binom{k + r - 1}{k} p^r (1-p)^k
P(X=k)=(kk+r−1)pr(1−p)k
Cumulative Distribution Function (CDF)：
F
(
X
=
k
)
=
∑
i
=
0
k
(
i
+
r
−
1
i
)
p
r
(
1
−
p
)
i
F(X = k) = sum_{i=0}^{k} binom{i + r - 1}{i} p^r (1-p)^i
F(X=k)=∑i=0k(ii+r−1)pr(1−p)i
2.5 Hypergeometric Distribution
definition: describes the process of sampling from a finite population without replacement.
n
n
n times, success
k
k
k times the probability.
formula：
X
∼
Hypergeom
(
N
,
K
,
n
)
X sim text{Hypergeom}(N, K, n)
X∼Hypergeom(N,K,n)
Probability Mass Function (PMF)：
P
(
X
=
k
)
=
(
K
k
)
(
N
−
K
n
−
k
)
(
N
n
)
P(X = k) = frac{binom{K}{k} binom{N-K}{n-k}}{binom{N}{n}}
P(X=k)=(nN)(kK)(n−kN−K)
Cumulative Distribution Function (CDF)：
F
(
X
=
k
)
=
∑
i
=
0
k
(
K
i
)
(
N
−
K
n
−
i
)
(
N
n
)
F(X = k) = sum_{i=0}^{k} frac{binom{K}{i} binom{N-K}{n-i}}{binom{N}{n}}
F(X=k)=∑i=0k(nN)(iK)(n−iN−K)
2.6 Poisson Distribution
definition：Describes what happens in a unit of time
k
k
k The probability of an event is λ, and the average rate of events is λ.
formula：
X
∼
Poisson
(
λ
)
X sim text{Poisson}(lambda)
X∼Poisson(λ)
Probability Mass Function (PMF)：
P
(
X
=
k
)
=
λ
k
e
−
λ
k
!
P(X = k) = frac{lambda^k e^{-lambda}}{k!}
P(X=k)=k!λke−λ
Cumulative Distribution Function (CDF)：
F
(
X
=
k
)
=
e
−
λ
∑
i
=
0
k
λ
i
i
!
F(X = k) = e^{-lambda} sum_{i=0}^{k} frac{lambda^i}{i!}
F(X=k)=e−λ∑i=0ki!λi
2.7 Relationship between them
Bernoulli Distributionis a specialBinomial Distribution,when
n
=
1
n = 1
n=1 hour.
Geometric distributionis the distribution of the number of trials required before the first success, which can be viewed asBinomial Distributionextension of.
Negative binomial distributionCan be seen asGeometric distributionA generalization of , used to describe the number of failures required before r successes.
Hypergeometric distributionSimilar toBinomial Distribution, but is applicable to finite populations and sampling without replacement.
Poisson distributionyesBinomial DistributionIn the limiting case,
n
n
n Very large and
p
p
p Very small, and
λ
=
n
p
lambda = np
λ=np keep constant.
Continuous Probability Distribution
3.1 Exponential Distribution
definition: Exponential distribution is a continuous probability distribution, often used to describe the time intervals between independent events.
formula：
X
∼
Exponential
(
λ
)
X sim text{Exponential}(lambda)
X∼Exponential(λ),in
λ
>
0
lambda>0
λ>0 is the rate parameter
Probability density function (PDF)：
f
(
x
;
λ
)
=
λ
e
−
λ
x
for
x
≥
0
f(x; lambda) = lambda e^{-lambda x} quad text{for } x geq 0
f(x;λ)=λe−λxfor x≥0
Cumulative Distribution Function (CDF)：
F
(
x
;
λ
)
=
1
−
e
−
λ
x
for
x
≥
0
F(x; lambda) = 1 - e^{-lambda x} quad text{for } x geq 0
F(x;λ)=1−e−λxfor x≥0
3.2 Gamma Distribution
definition: Describes the accumulation of waiting time and is a generalization of the exponential distribution and the χ² distribution.
formula：
X
∼
Gamma
(
k
,
θ
)
X sim text{Gamma}(k, theta)
X∼Gamma(k,θ),in
k
>
0
k>0
k>0 is the shape parameter,
θ
>
0
theta>0
θ>0 is the scale parameter
Probability density function (PDF)：
f
(
x
;
k
,
θ
)
=
x
k
−
1
e
−
x
/
θ
θ
k
Γ
(
k
)
for
x
≥
0
f(x; k, theta) = frac{x^{k-1}e^{-x/theta}}{theta^k Gamma(k)} quad text{for } x geq 0
f(x;k,θ)=θkΓ(k)xk−1e−x/θfor x≥0,in
Γ
(
k
)
Gamma(k)
Γ(k) is the gamma function
Cumulative Distribution Function (CDF)：
F
(
x
;
k
,
θ
)
=
γ
(
k
,
x
/
θ
)
Γ
(
k
)
F(x; k, theta) = frac{gamma(k, x/theta)}{Gamma(k)}
F(x;k,θ)=Γ(k)γ(k,x/θ),in
γ
(
k
,
x
/
θ
)
gamma(k, x/theta)
γ(k,x/θ) is the incomplete gamma function.
3.3 Normal Distribution
definition: Describes the distribution of the sum of a large number of independent random variables and is widely used in natural sciences and social sciences.
formula：
X
∼
N
(
μ
,
σ
2
)
X sim mathcal{N}(mu, sigma^2)
X∼N(μ,σ2),in,
μ
mu
μ is the mean,
σ
2
sigma^2
σ2 is the variance
Probability density function (PDF)：
f
(
x
;
μ
,
σ
2
)
=
1
2
π
σ
2
e
−
(
x
−
μ
)
2
2
σ
2
f(x; mu, sigma^2) = frac{1}{sqrt{2pisigma^2}} e^{-frac{(x-mu)^2}{2sigma^2}}
f(x;μ,σ2)=2πσ21e−2σ2(x−μ)2
Cumulative Distribution Function (CDF)：
F
(
x
;
μ
,
σ
2
)
=
1
2
[
1
+
erf
(
x
−
μ
2
σ
2
)
]
F(x; mu, sigma^2) = frac{1}{2}left[1 + operatorname{erf}left(frac{x-mu}{sqrt{2sigma^2}}right)right]
F(x;μ,σ2)=21[1+erf(2σ2x−μ)], where erf is the error function.
3.4 t-Distribution
definition: Used for hypothesis testing and confidence interval estimation in small sample cases.
formula：
X
∼
t
(
ν
)
X sim t(nu)
X∼t(ν),in,
ν
nu
ν It is the degree of freedom
Probability density function (PDF)：
f
(
t
;
ν
)
=
Γ
(
ν
+
1
2
)
ν
π
Γ
(
ν
2
)
(
1
+
t
2
ν
)
−
ν
+
1
2
f(t; nu) = frac{Gammaleft(frac{nu+1}{2}right)}{sqrt{nupi} Gammaleft(frac{nu}{2}right)} left(1 + frac{t^2}{nu}right)^{-frac{nu+1}{2}}
f(t;ν)=νπΓ(2ν)Γ(2ν+1)(1+νt2)−2ν+1
Cumulative Distribution Function (CDF)：
F
(
t
;
ν
)
=
1
2
+
t
Γ
(
ν
+
1
2
)
π
ν
Γ
(
ν
2
)
∫
0
t
(
1
+
u
2
ν
)
−
ν
+
1
2
d
u
F(t; nu) = frac{1}{2} + tfrac{Gammaleft(frac{nu+1}{2}right)}{sqrt{pi nu} Gammaleft(frac{nu}{2}right)} int_0^{t} left(1 + frac{u^2}{nu}right)^{-frac{nu+1}{2}} du
F(t;ν)=21+tπνΓ(2ν)Γ(2ν+1)∫0t(1+νu2)−2ν+1du
3.5 Chi-Square Distribution
definition: Commonly used in hypothesis testing and analysis of variance.
formula：
X
∼
χ
2
(
k
)
X sim chi^2(k)
X∼χ2(k),in
k
k
k It is the degree of freedom
Probability density function (PDF)：
f
(
x
;
k
)
=
1
2
k
/
2
Γ
(
k
/
2
)
x
k
/
2
−
1
e
−
x
/
2
for
x
≥
0
f(x; k) = frac{1}{2^{k/2} Gamma(k/2)} x^{k/2-1} e^{-x/2} quad text{for } x geq 0
f(x;k)=2k/2Γ(k/2)1xk/2−1e−x/2for x≥0
Cumulative Distribution Function (CDF)：
F
(
x
;
k
)
=
γ
(
k
2
,
x
2
)
Γ
(
k
2
)
F(x; k) = frac{gammaleft(frac{k}{2}, frac{x}{2}right)}{Gammaleft(frac{k}{2}right)}
F(x;k)=Γ(2k)γ(2k,2x)
3.6 F-Distribution
definition: Used to compare the variance of two samples.
formula：
X
∼
F
(
d
1
,
d
2
)
X sim F(d_1, d_2)
X∼F(d1,d2),in
d
1
d_1
d1 and
d
2
d_2
d2 It is the degree of freedom
Probability density function (PDF)：
f
(
x
;
d
1
,
d
2
)
=
(
d
1
x
)
d
1
d
2
d
2
(
d
1
x
+
d
2
)
d
1
+
d
2
x
B
(
d
1
2
,
d
2
2
)
for
x
≥
0
f(x; d_1, d_2) = frac{sqrt{frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1 + d_2}}}}{x Bleft(frac{d_1}{2}, frac{d_2}{2}right)} quad text{for } x geq 0
f(x;d1,d2)=xB(2d1,2d2)(d1x+d2)d1+d2(d1x)d1d2d2for x≥0,in
B
B
B is the beta function
Cumulative Distribution Function (CDF)：
F
(
x
;
d
1
,
d
2
)
=
I
d
1
x
d
1
x
+
d
2
(
d
1
2
,
d
2
2
)
F(x; d_1, d_2) = I_{frac{d_1 x}{d_1 x + d_2}}left(frac{d_1}{2}, frac{d_2}{2}right)
F(x;d1,d2)=Id1x+d2d1x(2d1,2d2),in
I
I
I is an incomplete beta function
3.7 Relationship between them
The chi-square distribution isSum of squares of a normal distribution.For example,
k
k
k The sum of squares of independent standard normal variables follows the degree of freedom
k
k
k The chi-square distribution of .
The t distribution isConstructed on the basis of standard normal distribution and chi-square distributionSpecifically, the t-distribution can be obtained by dividing a standard normal variate by the square root of its independent chi-squared distributed variate.
The F distribution isExtension of the ratio of two independent chi-squared distributed variablesThe F distribution is used to compare two variances and is constructed by taking the ratio of two chi-squared distributed variables.
