Raising the bar Chi-Square Distribution Archives

Dec 16

Ditch statistical tables if you have a computer

category: Agresti & Franklin, Agresti and Franklin, AP Stats and Business Stats, EC2020, EC2020 Elements of econometrics, Econometrics, ST104A, ST104B, Statistics 1, Statistics 2, The College Board 1 comment

You don't need statistical tables if you have Excel or Mathematica. Here I give the relevant Excel and Mathematica functions described in Chapter 14 of my book. You can save all the formulas in one spreadsheet or notebook and use it multiple times.

Cumulative Distribution Function of the Standard Normal Distribution

For a given real $z$ , the value of the distribution function of the standard normal is
$F(z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z}\exp (-t^{2}/2)dt.$

In Excel, use the formula =NORM.S.DIST(z,TRUE).

In Mathematica, enter CDF[NormalDistribution[0,1],z]

Probability Function of the Binomial Distribution

For given number of successes $x,$ number of trials $n$ and probability $p$ the probability is

$P(Binomial=x)=C_{x}^{n}p^{x}(1-p)^{n-x}$ .

In Excel, use the formula =BINOM.DIST(x,n,p,FALSE)

In Mathematica, enter PDF[BinomialDistribution[n,p],x]

Cumulative Binomial Probabilities

For a given cut-off value $x,$ number of trials $n$ and probability $p$ the cumulative probability is

$P(Binomial\leq x)=\sum_{t=0}^{x}C_{t}^{n}p^{t}(1-p)^{n-t}.$
In Excel, use the formula =BINOM.DIST(x,n,p,TRUE).

In Mathematica, enter CDF[BinomialDistribution[n,p],x]

Values of the exponential function $e^{-\lambda}$

In Excel, use the formula =EXP(-lambda)

In Mathematica, enter Exp[-lambda]

Individual Poisson Probabilities

For given number of successes $x$ and arrival rate $\lambda$ the probability is

$P(Poisson=x)=\frac{e^{-\lambda }\lambda^{x}}{x!}.$
In Excel, use the formula =POISSON.DIST(x,lambda,FALSE)

In Mathematica, enter PDF[PoissonDistribution[lambda],x]

Cumulative Poisson Probabilities

For given cut-off $x$ and arrival rate $\lambda$ the cumulative probability is

$P(Poisson\leq x)=\sum_{t=0}^{x}\frac{e^{-\lambda }\lambda ^{t}}{t!}.$
In Excel, use the formula =POISSON.DIST(x,lambda,TRUE)

In Mathematica, enter CDF[PoissonDistribution[lambda],x]

Cutoff Points of the Chi-Square Distribution Function

For given probability of the right tail $\alpha$ and degrees of freedom $\nu$ , the cut-off value (critical value) $\chi _{\nu,\alpha }^{2}$ is a solution of the equation
$P(\chi _{\nu}^{2}>\chi _{\nu,\alpha }^{2})=\alpha .$
In Excel, use the formula =CHISQ.INV.RT(alpha,v)

In Mathematica, enter InverseCDF[ChiSquareDistribution[v],1-alpha]

Cutoff Points for the Student’s t Distribution

For given probability of the right tail $\alpha$ and degrees of freedom $\nu$ , the cut-off value $t_{\nu,\alpha }$ is a solution of the equation $P(t_{\nu}>t_{\nu,\alpha })=\alpha$ .
In Excel, use the formula =T.INV(1-alpha,v)

In Mathematica, enter InverseCDF[StudentTDistribution[v],1-alpha]

Cutoff Points for the F Distribution

For given probability of the right tail $\alpha$ , degrees of freedom $v_1$ (numerator) and $v_2$ (denominator), the cut-off value $F_{v_1,v_2,\alpha }$ is a solution of the equation $P(F_{v_1,v_2}>F_{v_1,v_2,\alpha })=\alpha$ .

In Excel, use the formula =F.INV.RT(alpha,v1,v2)

In Mathematica, enter InverseCDF[FRatioDistribution[v1,v2],1-alpha]

Tags: Binomial Distribution, Chi-Square Distribution, Cumulative Binomial Probabilities, Cumulative Distribution Function, Cumulative Poisson Probabilities, Excel, exponential function, F distribution, Mathematica, Poisson distribution, Poisson Probabilities, Probability Function, Standard Normal Distribution, statistical tables, Student's t distribution

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