Formulae - Statistics I
This is the formulae for the course, that you will be given for the midterm and exam.
Index Numbers
- \(i_{t|0}=\frac{x_t}{x_0}\times 100\) with \(x_t, x_0>0\)
- \(\delta_{t,0}=\frac{x_t-x_0}{x_0}\)
- \(\delta_{t,0}=i_{t|0}-1\)
- \(r_{t|0}=\left(i_{t|0}\right)^{1/k}-1\)
- \(LPI_{t|0}=\frac{\sum p_t^k q_0^k}{\sum p_0^k q_0^k}\)
- \(LQI_{t|0}=\frac{\sum p_0^k q_t^k}{\sum p_0^k q_0^k}\)
- \(PPI_{t|0}=\frac{\sum p_t^k q_t^k}{\sum p_0^k q_t^k}\)
- \(PQI_{t|0}=\frac{\sum p_t^k q_t^k}{\sum p_t^k q_0^k}\)
- \(FPI_{t|0}=\sqrt{LPI_{t|0}PPI_{t|0}}\)
- \(FQI_{t|0}=\sqrt{LQI_{t|0}PQI_{t|0}}\)
Probability Theory
Conditional probability
- \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) with \(P(B)>0\)
Total Probability Theorem
- \(P(B)=\sum_{i=1}^n P(A_i)P(B|A_i)\)
Bayes Theorem
- \(P(A_i|B)=\frac{P(A_i)P(B|A_i)}{\sum_{i=1}^n P(A_i)P(B|A_i)}\)
Random variables
Expected Value
- \(\mu_X \equiv E[X]\)
- \(\mu_X = \sum x_i f(x_i)\)
- \(\mu_X = \int x f(x) dx\)
Variance
- \(\sigma_X^2 = Var(X) = V[X]\)
- \(\sigma_X^2 = E[(X-\mu_X)^2]=E[X^2]-\mu_X^2\)
Discrete Joint Random Variables
Joint density function
- \(f_{XY}(x,y)=P(X=x_i,Y=y_i)=p_{ij}\) with \(p_i=\sum_j p_{ij}\) and \(p_j=\sum_i p_{ij}\)
Expected Value
- \(E[g(X,Y)]=\sum_i\sum_j g(x_i,y_j)P(X=x_i,Y=y_j)\)
Variance, Covariance, Correlation
- \(cov(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-E[X]E[Y]\)
- \(V[X\pm Y]=V[X]+V[Y]\pm 2cov(X,Y)\)
- \(cov(a+bX,c+dY)=bdcov(X,Y)\) with \(a,b,c,d\in\mathbb{R}\)
- \(\rho = \frac{cov(X,Y)}{\sqrt{V[X]V[Y]}}\) with \(\sigma_X, \sigma_Y > 0\)
Discrete Probabilistic Models
Binomial Distribution
- \(X\sim Bin(n,p)\)
- \(P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}\) with \(x=0,1...,n\)
- \(E[X]=np\)
- \(V[X]=npq\)
- \(q=1-p\)
Hypergeometric Distribution
- \(X\sim Hypergeometric(N,M,n)\) with \(p=\frac{M}{N}\), \(q=1-p\)
- \(P(X=x)=\frac{\binom{n}{x}\binom{N-M}{n-x}}{\binom{N}{n}}\) with \(max(0,M+n-N)\leq x \leq min(M,n)\)
- \(E[X]=np\)
- \(V[X]=npq\frac{N-n}{N-1}\)
Geometric Distribution
- \(X\sim Geo(p)\) \(0\leq p \leq 1\)
- \(P(X=x)=p(1-p)^{x-1}\) with \(x=1,2,...\)
- \(E[x]=\frac{1}{p}\)
- \(V[X]=\frac{1-p}{p^2}\)
- \(F(x)=\begin{cases}0 & x<1 \\ 1-(1-p)^k & k\leq x < k+1\quad k=1,2..\end{cases}\)
Poisson Distribution
- \(X\sim Poi(\lambda)\) with \(\lambda>0\)
- \(P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}\) with \(x=0,1,2...\)
- \(E[X]=V[X]=\lambda\)
Continuous Probabilistic Models
Uniform Distribution
- \(X\sim Unif(a,b)\) with \(a,b\in\mathbb{R}\) and \(a<b\)
- \(F(x)=\begin{cases}0 & x< a\\ \frac{x-a}{b-a} & a\leq x < b\\ 1 & x\geq b\end{cases}\)
- \(E[X]=\frac{a+b}{2}\)
- \(V[X]=\frac{(b-a)^2}{12}\)
Exponential Distribution
- \(X\sim Exp(\lambda)\) with \(\lambda>0\)
- \(F(x)=\begin{cases}0 & x\leq 0 \\ 1-e^{-\lambda x} & x>0\end{cases}\)
- \(E[x]=\frac{1}{\lambda}\)
- \(V[X]=\frac{1}{\lambda^2}\)
Normal Distribution
- \(X\sim \mathcal{N}(\mu,\sigma)\)
- \(\mu\in\mathbb{R}\), \(\sigma>0\)
- \(E[X]=\mu\)
- \(V[X]=\sigma^2\)
- \(Z=\frac{X-\mu}{\sigma}\sim \mathcal{N}(0,1)\)
- \(E[Z]=0\)
- \(V[Z]=1\)
- \(\Phi(z)=P(Z\leq z)\)
- \(\Phi(-z)=1-\Phi(z)\)