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STATISTICS (CONCISE (DISTRIBUTION (Normal variable x (X~N(μ,σ^2 ), E(x)=μ,…

- - - - Discrete
        
        Frequency Distribution - a list, table or graph that displays the frequency of various outcomes in a sample.
        Example :
        
        Vertical Line Graph
        
        line - represents frequency
        
        distinct lines - shows discrete nature of variables
        
        Bar Chart
        
        bar - represents frequency
        
        Mode - value that occurs most often, most popular,a variable with highest frequency
        
        Examples of discrete raw data :
        Number of siblings of students in S18D
        1,2,4,5,8,1,3,4,7,4,2,5,4,6,3,4
        
        Number of cars passing a checkpoint in 30 minutes
        
        The shoe sizes of children in a class
        
        Why raw?
        
        They have not been ordered in any way
        
        Characteristic of discrete data
        
        Can take only exact values
        
        Calculation of mean, standard deviation and variance
        
        VARIANCE
        
        STANDARD DEVIATION
        
        MEAN
        
        x= sum of all x
        n=number of sample
      - Continuous
        
        Examples of continuous raw data :
        
        The speed of a vehicle as it passes a checkpoint
        
        The mass of a cooking apple
        
        The time taken by a volunteer to perform a task.
        
        Characteristic of continuous data
        
        cannot take exact values but can give only within specified range/ measured to a specified degree of accuracy
        
        Frequency Distribution
        Step 1 : Group the information into classes or intervals
        
        Way 1 to write the
        same set of intervals :
        Height (cm)
        119.5\ \le h\ <\ 124.5\
        124.5 < h < 129.5
        129.5 < h < 134.5
        134.5 < h < 139.5
        139.5 < h < 144.5
        
        Way 2 to write the
        same set of intervals :
        Height (cm)
        119.5 - 124.5
        124.5 - 129.5
        129.5 - 134.5
        134.5 - 139.5
        139.5 - 144.5
        
        The values 119.5, 124.5, 129.5 are called class boundaries or interval boundaries
        
        The upper class boundary of one interval is the lower class boundary of the next interval
        
        Width of an interval = Upper class boundary - Lower class boundary
        
        Way 3 to write the
        same set of intervals :
        Height (to the nearest cm)
        120 -124
        125 - 129
        130 - 134
        135 - 139
        140 - 144
        
        *
        Calculation of mean, Standard Deviation and variance using formula.
        
        Standard Deviation
        
        x= is midpoint of the class interval
        
        Variance
        
        Mean
    - - Venn Diagram
        
        \[A\,\cup\,B\,(A\,or\,B\,or\,both)\]
        \[P(A\,\cup\,B)=P(A)+P(B)-P(A\,\cap\,B)\]
        
        \[A\,\cap\,B\,(both\,A\,and\,B)\]
        
        For independent events, \[P(A\,\cap\,B)=P(A)\,\times\,P(B)\]
        
        \[Conditional\,Probability\]
        
        For independent events, P(A|B)=P(A)
        
        \[Complementary\,event,\,\bar{A}\,or\,A'\]
        P(A) + P(A') = 1 \[Exhaustive\,Event\]
        \[P(A\,\cup\,A')=1\]
      - Tree Diagram
        
        Bayes theorem
        
        Reverse condition
        -We know P(B|A) but we want P(A|B)
        -Same formula as conditional probability
    - - n unlike objects \[n!\]
      - n unlike objects with p alike object \[\frac{n!}{p!}\]
      - n unlike objects in a ring \[(n-1)!\]
      - Permutation
        \[^{n}P_{r} = \frac{n!}{(n-r)!}\]
      - Combination
        \[^{n}C_{r}\,or\,_{n}C_{r}=\frac{n!}{n!(n-r)!}\]
    - - why use pdf, for a continuous random variable, the probability can only be in a range (area under the curve)
        
        For a continuous variable it does not matter whether you use strict inequalities (a < × < b) or inclusive inequalities (a ≤ × ≤ b).
      - Properties
        
        A probability density function will often have different rules over different domains. If a probability density function is only defined over a particular domain you may assume that it is zero everywhere else.
      - Applications
        
        Median
        
        Expectation (mean)
        
        Mode value of x at the maximum value of f (x).
        
        It will be not necessary where dt=0 for the x to be the mode of the function.
  - - - X~N(μ,σ^2 )
      - E(x)=μ
      - Var(x)=σ^2
    - - Z~N(0,1)
      - E(Z)=0
      - Var(Z)=1
    - - Binomial Distribution
        
        Expectation of X
        E(X) = np
        
        Variance of X
        Var(X) = npq
        
        Probability Density Function
        
        Mode of X
        near with the mean
        
        Definition
        X = number of successful outcomes in n in dependent trials.
      - Poisson Distribution
        
        Expectation of X
        E(X) = lambda
        
        Variance of X
        Var(X) = lambda
        
        Probability Density Function
        
        Mode of X
        If lambda is an integer, the mode is lambda.
        If lambda is non integer, the mode is lambda and lambda-1.
        
        Definition
        X = number of occurrences in the given interval / time.
- - - - Random variable X is an unbiased estimator of µ if
        \[E(X)=\mu\]
      - Unbiased estimator of \({\sigma^2}\)
        \[S_{n-1}^2=\frac{n}{n-1}S_{n}\]
      - If a population has a parameter a then the sample statistic
        Â is an unbiased estimator of a if E(Â) = a.
        
        an accurate statistic that's used to approximate a population parameter
      - Important Key Notes
        
        An estimator is unbiased if its expected value equals the population mean of X
        
        The most efficient estimator has the smallest variance
      - Unbiased estimator of σ
        \[S_{n-1}=\sqrt\frac{n}{n-1}S_{n}\]
    - - Mean
        
        discrete random variable
        \[E(g(X))=\sum_{i}g(x_{i})p_{i}\]
        
        Continuous random variable with probability density function f(x)\[E(g(X))=\int g(x)f(x)dx\]\[E(X)=\int xf(x)dx\]
      - Mean and variance for sample
        
        For Sample mean
        \[E(\bar{X})=E(X)\]\[Var(\bar{X})=\frac{Var(X)}{n}\]
        
        For sample sum\[E(\sum_{i=1}^{n}X_{i})=nE(X)\]\[Var(\sum_{i=1}^{n}X_{i})=nVar(X)\]
      - General
        
        \[E(a_{1}X_{1}\pm a_{2}X_{2}\pm a_{3})=a_{1}E(X_{1})\pm a_{2}E(X_{2})\pm a_{3} \]\[Var(a_{1}X_{1}\pm a_{2}X_{2}\pm a_{3})=a{_{1}}^{2}Var(X_{1})+ a{_{2}}^{2}Var(X_{2})\]
        
        If X and Y are independent random variables then:
        \[E(XY)=E(X)E(Y)\]
      - Linear combinations of normal variables
        
        If X and Y are random variables following a normal distribution and Z=aX+bY+c then Z also follows a normal distribution.
      - Central limit theorem
        
        For any distribution, if \(E(X)=\mu\), \(Var(X)=\sigma ^{2}\) and \(n\geqslant 30\), then the approximate distributions of the sum and the mean are given by:
        \[\sum_{i=1}^{n}X_{i}\sim N(n\mu,n\sigma ^{2})\]\[\bar{X}\sim N(\mu,\frac{\sigma^{2}}{n})\]
    - - t-distribution:
        \[\frac{\bar{X}-\mu}{S_{n-1}/{\sqrt n}}\sim t_{n-1}\]
      - General formula for known variance:
        \[\bar{x}\pm z\frac{\sigma}{\sqrt n}\]
      - Width of confidence interval:
        \[2z\frac{\sigma}{\sqrt n}\]
      - Confidence interval for a mean with unknown variance:
        \[\bar{x}\pm t\frac{S_{n-1}}{\sqrt n}\]
    - - General procedure for hypothesis testing:
        
        Define variables
        
        State hypothesis \[H_0 \ ,\ H_1\]
        
        State significance level
        
        Decide which type of test statistic
        
        State distribution assuming that null hypothesis is true
        
        Calculate test statistic
        
        Conclusion (state whether the statistic is sufficiently unlikely by using p-value or critical region)
      - One-tailed and two-tailed test
      - Error in hypothesis testing:
  - - - Meaning
        
        The number of trials up to and including the first success, x
        
        Parameter
        
        X ~ Geo(p)
      - PDF
        
        P(X = x) = q^(x-1)p, where q = 1 - p
        \[P(X=x)=q^{x-1}p\]
      - Expectation
        
        E(X) = 1/p
        
        Variance
        
        Var(X) = q/(p^2), where q = 1 – p
        
        Mode
        
        when x=1
    - - The cumulative distribution function gives the probability of the random variable taking a value less than or equal to x.
      - Formula
        
        For a discrete distribution with probability mass function \(P(X = x)\):
        \[P(X\leq x)=\sum_{i=-\infty }^{i=x}p_{i}\]
        
        For a continuous distribution with pdf \(f(x)\):
        \[P(X\leq x)= F(x)=\int_{-\infty }^{x}f(x)dx\]
      - \[f(x)=\frac{d}{dx}F(x)\]
      - If X has cdf \(F(x)\) for \(a< x< b\) and \(Y=g(X)\) (where \(g(X)\) is a 1-to-1 function) then the pdf of Y, \(h(y)\), is given by:
        1) Relating \(H(y)\) to \(F(g^{-1}(y))\) by rearranging the inequality in \(P(Y\leq X)=P(g(X)\leq y)\).
        2) Differentiating \(H(y)\) with respect to y.
        3) Writing the domain of \(h(y)\) by solving the inequality \(a< g^{-1}(y)< b\)
    - - PDF
        \[^{x-1}C_{r-1}\,=p^{r} q^{x-r}\]
        where x= r, r+1, r+2, ...
      - X= Number of trials up to and including r successes
        Parameter
        X~ NB(r,p)
      - Expectation
        E(X)= r/p
        Variance
        Var(X)= rq/(p^2)
    - - The probability generating function of the discrete random variable X is given by :
      - Properties
        
        Gx(t)=E(t^x)
        
        For Any PGF : G(1)=1
        
        Mean and Variance
        
        G'(1)=E(x)
        
        Var(x)=G''(1)-G'(1)-(G'(1))^2
        
        G(t)=P(x=0)+P(x=1)t+P(x=2)t^2+...
        
        G(0)=P(x=0)
        
        G'(t)=P(x=1)+P(x=2)2t+P(x=3)3t^2+...
        
        G'(0)=P(x=1)
        
        G''(t)=P(x=2)2+P(x=3)6t+P(x=4)12t^2+...
        
        G''(0)=P(x=2)2P
        
        When we have two variables :
        
        If Z = X + Y , where x and y are independent
        
        Gz(t)=Gx(t) x Gy(t)
        
        Gz(t)=E(t^z)=E(t^X+Y)=E(t^X x t^Y)=E(t^X) x E(T^Y)=Gx(t) x Gy(t)
        
        In General:
        
        P(X=n)=(1/n!)G^(n)(0)
      - G(t)=(SIGMA)P(X=x)t^x