Mathematical Stat Discrete Variable Binomial distribution
Definition
The typical example is binomial distribution. Before we deal with the binomial distribution, we first have to define the bernoulli distribution.
X ( s u c c e s s ) = 1 , X ( f a i l u r e ) = 0 p ( x ) = p x ( 1 − p ) 1 − x , x = 0 , 1 X(success)=1 , \quad X(failure)=0 \\
p(x)=p^x(1-p)^{1-x}, x=0,1 X ( s u ccess ) = 1 , X ( f ai l u re ) = 0 p ( x ) = p x ( 1 − p ) 1 − x , x = 0 , 1 A sequence of bernoulli trial makes binomial distribution.
E ( X ) = p , V a r ( X ) = p ( 1 − p ) , M ( t ) = p e t + q E(X)=p, \; Var(X)=p(1-p), \; M(t)=pe^t+q E ( X ) = p , Va r ( X ) = p ( 1 − p ) , M ( t ) = p e t + q
When X is the number of successes in n independent Bernoulli trials, we say that X follows Binomial distribution .
Properties
f ( X ) = ( n x ) p x ( 1 − p ) n − x E ( X ) = n p , V a r ( X ) = n p ( 1 − p ) M ( t ) = ( p e t + q ) n , − ∞ < t < ∞ X ≡ Z 1 + ⋯ + Z n , Z i ∼ i i d B e r n o u l l i ( p ) f(X)=\binom n x p^x(1-p)^{n-x} \\
E(X)=np , \; Var(X)=np(1-p) \\
M(t)=(pe^t+q)^n, \; -\infty<t<\infty \\
X \equiv Z_1 +\cdots +Z_n , \; Z_i \sim iid \; Bernoulli(p) f ( X ) = ( x n ) p x ( 1 − p ) n − x E ( X ) = n p , Va r ( X ) = n p ( 1 − p ) M ( t ) = ( p e t + q ) n , − ∞ < t < ∞ X ≡ Z 1 + ⋯ + Z n , Z i ∼ ii d B er n o u ll i ( p ) Example
Let the winning rate of a gambler be 2 over 3. X is the number of winning for 3 games.
p ( x ) = 3 C x ( 2 3 ) x ( 1 3 ) 3 − x , x = 0 , 1 , 2 , 3 X ∼ B ( 3 , 2 3 ) p(x)=_3C_x(\dfrac{2}{3})^x(\dfrac{1}{3})^{3-x}, x=0,1,2,3 \\
X \sim B(3,\frac{2}{3}) p ( x ) = 3 C x ( 3 2 ) x ( 3 1 ) 3 − x , x = 0 , 1 , 2 , 3 X ∼ B ( 3 , 3 2 ) Negative Binomial distribution
Definition
Y Y Y r t h r_{th} r t h
p ( y ) = y + r − 1 C r − 1 p r − 1 ( 1 − p ) y ∗ p Y ∼ N B ( r , p ) p(y)=_{y+r-1}C_{r-1} p^{r-1}(1-p)^y*p \\
Y \sim NB(r,p) p ( y ) = y + r − 1 C r − 1 p r − 1 ( 1 − p ) y ∗ p Y ∼ NB ( r , p )
Poisson distribution
X ∼ P o i s s o n ( λ ) , λ > 0 f ( x ; λ ) = λ x e − λ x ! , x = 0 , 1 , 2 , ⋯ P ( k e v e n t s i n t i m e p e r i o d ) = e − e v e n t s t i m e ∗ t i m e p e r i o d ( e v e n t s t i m e ∗ t i m e p e r i o d ) k k ! X \sim Poisson(\lambda), \quad \lambda >0 \\
f(x;\lambda)=\dfrac{\lambda^xe^{-\lambda}}{x!}, \quad x=0,1,2,\cdots \\
P(k \; events \; in \; time \; period)=e^{-\frac{events}{time}*time\;period}\dfrac{(\frac{events}{time}*time \; period)^k}{k!} X ∼ P o i sso n ( λ ) , λ > 0 f ( x ; λ ) = x ! λ x e − λ , x = 0 , 1 , 2 , ⋯ P ( k e v e n t s in t im e p er i o d ) = e − t im e e v e n t s ∗ t im e p er i o d k ! ( t im e e v e n t s ∗ t im e p er i o d ) k λ \lambda λ
https://towardsdatascience.com/the-poson-distribution-and-poisson-process-explained-4e2cb17d459
Properties
E ( X ) = V a r ( X ) = λ M ( t ) = e m ( e t − 1 ) Y = X 1 + ⋯ X n ∼ P o i s s o n ( λ 1 + ⋯ λ n ) E(X)=Var(X)=\lambda \\
M(t)=e^{m(e^t-1)} \\
Y = X_1 +\cdots X_n \sim Poisson(\lambda_1+\cdots \lambda_n) E ( X ) = Va r ( X ) = λ M ( t ) = e m ( e t − 1 ) Y = X 1 + ⋯ X n ∼ P o i sso n ( λ 1 + ⋯ λ n ) E ( X ) = ∑ x = 0 n x λ x e − λ x ! = λ E(X)=\sum^n_{x=0} x\dfrac{\lambda^x e^{-\lambda}}{x!} = \lambda E ( X ) = x = 0 ∑ n x x ! λ x e − λ = λ M X ( t ) = E ( e t X ) = ∑ x = 0 ∞ e t x × λ x e − x x ! = e − λ ∑ x = 0 ∞ ( λ e t ) x x ! = e λ ( e t − 1 ) \begin{split}
M_X(t) &= E(e^{tX}) \\
& = \sum^{\infty}_{x=0} e^{tx} \times \dfrac{\lambda^x e^{-x}}{x!} \\
&= e^{-\lambda} \sum^{\infty}_{x=0}\dfrac{(\lambda e^t)^x}{x!} \\
& = e^{\lambda (e^t-1)}
\end{split} M X ( t ) = E ( e tX ) = x = 0 ∑ ∞ e t x × x ! λ x e − x = e − λ x = 0 ∑ ∞ x ! ( λ e t ) x = e λ ( e t − 1 ) Poisson & Binomial
In a rare event situation(n is so large and p is so small in binomial distribution), binomial distribution asymptotically can be a poisson distribution.
lim n → ∞ n ! n x ( n − x ) ! = 1 , lim n → ∞ ( 1 − u n ) n = e − u , lim n → ∞ ( 1 − u n ) − x = 1 \lim_{n \to \infty}\dfrac{n!}{n^x(n-x)!}=1, \; \lim_{n \to \infty}(1-\frac{u}{n})^n=e^{-u}, \; \lim_{n \to \infty}(1-\frac{u}{n})^{-x}=1
n → ∞ lim n x ( n − x )! n ! = 1 , n → ∞ lim ( 1 − n u ) n = e − u , n → ∞ lim ( 1 − n u ) − x = 1 lim n → ∞ b ( x ; n , p ) = lim n → ∞ n ! x ! ( n − x ) ! p x ( 1 − p ) n − x = lim n → ∞ n x x ! n ! n x ( n − x ) ! p x ( 1 − p ) n − x = lim n → ∞ ( n p ) x ( 1 − p ) n − x x ! = lim n → ∞ ( n p ) x ( 1 − n p n ) n − x x ! = λ x e − λ x ! \begin{split}
\lim_{n \to \infty}b(x;n,p) &= \lim_{n \to \infty}\dfrac{n!}{x!(n-x)!}p^x(1-p)^{n-x} \\
&= \lim_{n \to \infty} \dfrac{n^x}{x!} \dfrac{n!}{n^x(n-x)!} p^x(1-p)^{n-x} \\
&= \lim_{n \to \infty} \dfrac{(np)^x(1-p)^{n-x}}{x!} \\
&= \lim_{n \to \infty} \dfrac{(np)^x(1-\frac{np}{n})^{n-x}}{x!} \\
&= \dfrac{\lambda^xe^{-\lambda}}{x!}
\end{split} n → ∞ lim b ( x ; n , p ) = n → ∞ lim x ! ( n − x )! n ! p x ( 1 − p ) n − x = n → ∞ lim x ! n x n x ( n − x )! n ! p x ( 1 − p ) n − x = n → ∞ lim x ! ( n p ) x ( 1 − p ) n − x = n → ∞ lim x ! ( n p ) x ( 1 − n n p ) n − x = x ! λ x e − λ
Exercises
