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Statistics: probability, estimation and inference

Connect probability models to sampling, parameter estimation, intervals, regression and reproducible interpretation.

Subject library · 51 guides · derivations & worked examples

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Probability and moments

Definitions & inputs. X random variable,p probability mass,density f,μ mean,σ² variance.

  1. Weighted outcomes define expectation and spread.

    E[X]=∑xxp(x),Var⁡(X)=E[X2]−E[X]2E[X]=\sum_x xp(x),\quad\operatorname{Var}(X)=E[X^2]-E[X]^2

  2. Transform the random variable and center its fluctuations.

    E[aX+b]=aμ+b,Var⁡(aX+b)=a2σ2E[aX+b]=a\mu+b,\quad\operatorname{Var}(aX+b)=a^2\sigma^2

  3. Cross terms matter unless covariance is zero.

    Var⁡(X+Y)=σX2+σY2+2Cov⁡(X,Y)\operatorname{Var}(X+Y)=\sigma_X^2+\sigma_Y^2+2\operatorname{Cov}(X,Y)

Interpretation. Zero correlation is weaker than independence; probabilities and densities have different units.

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2. Sampling and estimation

Definitions & inputs. xi independent observations,n count,xbar sample mean,s² sample variance.

  1. Fitting the sample mean consumes one degree of freedom, giving the unbiased variance denominator.

    xˉ=1n∑ixi,s2=1n−1∑i(xi−xˉ)2\bar x=\frac1n\sum_i x_i,\quad s^2=\frac1{n-1}\sum_i(x_i-\bar x)^2

  2. Independent sample variances add and the mean divides their sum byn².

    SE(xˉ)=σ/nSE(\bar x)=\sigma/\sqrt n

  3. Under normal sampling the statistic follows a t distribution withn−1 degrees of freedom under the stated null.

    t=(xˉ−μ0)/(s/n)t=(\bar x-\mu_0)/(s/\sqrt n)

Interpretation. More observations reduce random sampling error but do not automatically remove selection bias or measurement offset.

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3. Intervals and hypothesis tests

Definitions & inputs. α error rate,z or t quantile,μ unknown population mean.

  1. A repeated-sampling normal interval has nominal coverage under its model.

    CI=xˉ±z1−α/2σ/nCI=\bar x\pm z_{1-\alpha/2}\sigma/\sqrt n

  2. Estimating variance broadens the critical value, especially for smalln.

    CIt=xˉ±tn−1,1−α/2s/nCI_t=\bar x\pm t_{n-1,1-\alpha/2}s/\sqrt n

  3. A p-value is conditional on the null model; it is not the probability that the null is true.

    p=PH0(test statistic at least as extreme as observed)p=P_{H_0}(\text{test statistic at least as extreme as observed})

Interpretation. Report effect sizes, uncertainty, selection of analyses and multiple-testing control, not only a significance label.

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4. Regression and Bayesian updating

Definitions & inputs. y observations,X design matrix,β coefficients,ε errors,θ parameter,D data.

  1. Differentiate squared residual loss and solve the normal equations; QR or SVD is preferable numerically.

    β^=(XTX)−1XTy\widehat\beta=(X^TX)^{-1}X^Ty

  2. Homoscedastic independent errors produce the familiar covariance; use other estimators when assumptions fail.

    Cov⁡(β^)=σ2(XTX)−1\operatorname{Cov}(\widehat\beta)=\sigma^2(X^TX)^{-1}

  3. Bayes’ theorem updates a prior using the likelihood.

    p(θ∣D)=p(D∣θ)p(θ)/p(D)p(\theta|D)=p(D|\theta)p(\theta)/p(D)

Interpretation. Confidence, prediction and credible intervals answer different questions; correlation or a fitted slope alone does not establish causation.

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Graphical worked example

Standard normal density; probabilities are areas under the curve, not density values at a point. X axis: Standardized value z (dimensionless). Y axis: Standard normal probability density.
Standard normal density; probabilities are areas under the curve, not density values at a point. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Mean

Definitions & inputs. Data2,4,6.

  1. Choose the governing model and isolate the requested quantity.

    xˉ=(2+4+6)/3\bar x=(2+4+6)/3

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    12/312/3

  3. Evaluate the expression; the result uses the units shown.

    Result=4 \mathrm{Result}=4\ {}

Interpretation. Arithmetic mean.

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Example 02. Sample variance

Definitions & inputs. Same data.

  1. Choose the governing model and isolate the requested quantity.

    s2=[(2−4)2+(4−4)2+(6−4)2]/2s^2=[(2-4)^2+(4-4)^2+(6-4)^2]/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    8/28/2

  3. Evaluate the expression; the result uses the units shown.

    Result=4 \mathrm{Result}=4\ {}

Interpretation. Usesn−1.

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Example 03. Sample standard deviation

Definitions & inputs. Variance4.

  1. Choose the governing model and isolate the requested quantity.

    s=s2s=\sqrt{s^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4\sqrt4

  3. Evaluate the expression; the result uses the units shown.

    Result=2 \mathrm{Result}=2\ {}

Interpretation. Same units as data.

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Example 04. Standard error

Definitions & inputs. σ2,n100.

  1. Choose the governing model and isolate the requested quantity.

    SE=σ/nSE=\sigma/\sqrt n

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2/102/10

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2 \mathrm{Result}=0.2\ {}

Interpretation. Independent samples.

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Example 05. Known variance95percent half width

Definitions & inputs. SE.2,z1.959964.

  1. Choose the governing model and isolate the requested quantity.

    h=zSEh=zSE

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1.959964(.2)1.959964(.2)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.3919928 \mathrm{Result}=0.3919928\ {}

Interpretation. Normal interval.

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Example 06. Normal z score

Definitions & inputs. x14,μ10,σ2.

  1. Choose the governing model and isolate the requested quantity.

    z=(x−μ)/σz=(x-\mu)/\sigma

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (14−10)/2(14-10)/2

  3. Evaluate the expression; the result uses the units shown.

    Result=2 \mathrm{Result}=2\ {}

Interpretation. Standardization.

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Example 07. Bernoulli variance

Definitions & inputs. p.3.

  1. Choose the governing model and isolate the requested quantity.

    V=p(1−p)V=p(1-p)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .3(.7).3(.7)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.21 \mathrm{Result}=0.21\ {}

Interpretation. Binary outcome.

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Example 08. Binomial mean

Definitions & inputs. n20,p.3.

  1. Choose the governing model and isolate the requested quantity.

    E[X]=npE[X]=np

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    20(.3)20(.3)

  3. Evaluate the expression; the result uses the units shown.

    Result=6 \mathrm{Result}=6\ {}

Interpretation. Independent identical trials.

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Example 09. Binomial variance

Definitions & inputs. Same.

  1. Choose the governing model and isolate the requested quantity.

    V=np(1−p)V=np(1-p)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    20(.3)(.7)20(.3)(.7)

  3. Evaluate the expression; the result uses the units shown.

    Result=4.2 \mathrm{Result}=4.2\ {}

Interpretation. Count variance.

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Example 10. Zero successes

Definitions & inputs. n5,p.2.

  1. Choose the governing model and isolate the requested quantity.

    P(X=0)=(1−p)nP(X=0)=(1-p)^n

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .85.8^5

  3. Evaluate the expression; the result uses the units shown.

    Result=0.32768 \mathrm{Result}=0.32768\ {}

Interpretation. Binomial model.

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Example 11. Poisson zero count

Definitions & inputs. λ3.

  1. Choose the governing model and isolate the requested quantity.

    P(0)=e−λP(0)=e^{-\lambda}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e−3e^{-3}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.04978707 \mathrm{Result}=0.04978707\ {}

Interpretation. E qui dispersed Poisson count.

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Example 12. Exponential mean

Definitions & inputs. Rate.2/s.

  1. Choose the governing model and isolate the requested quantity.

    E[T]=1/λE[T]=1/\lambda

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/.21/.2

  3. Evaluate the expression; the result uses the units shown.

    Result=5 s\mathrm{Result}=5\ \mathrm s

Interpretation. Memory less waiting time.

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Example 13. Uniform variance

Definitions & inputs. Uniform0to2.

  1. Choose the governing model and isolate the requested quantity.

    V=(b−a)2/12V=(b-a)^2/12

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4/124/12

  3. Evaluate the expression; the result uses the units shown.

    Result=0.3333333 \mathrm{Result}=0.3333333\ {}

Interpretation. Continuous distribution.

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Example 14. Independent sum variance

Definitions & inputs. Variances4and9.

  1. Choose the governing model and isolate the requested quantity.

    V=V1+V2V=V_1+V_2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4+94+9

  3. Evaluate the expression; the result uses the units shown.

    Result=13 \mathrm{Result}=13\ {}

Interpretation. Zero covariance.

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Example 15. Correlated sum variance

Definitions & inputs. σ12,σ23,correlation.5.

  1. Choose the governing model and isolate the requested quantity.

    V=4+9+2ρσ1σ2V=4+9+2\rho\sigma_1\sigma_2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    13+2(.5)(2)(3)13+2(.5)(2)(3)

  3. Evaluate the expression; the result uses the units shown.

    Result=19 \mathrm{Result}=19\ {}

Interpretation. Co variance must be in units of product.

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Example 16. Slope

Definitions & inputs. Exact points(0,1),(1,3),(2,5).

  1. Choose the governing model and isolate the requested quantity.

    b=(5−1)/(2−0)b=(5-1)/(2-0)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4/24/2

  3. Evaluate the expression; the result uses the units shown.

    Result=2 \mathrm{Result}=2\ {}

Interpretation. Perfect linear illustration.

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Example 17. Intercept

Definitions & inputs. y1atx0,slope2.

  1. Choose the governing model and isolate the requested quantity.

    a=y−bxa=y-bx

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1−2(0)1-2(0)

  3. Evaluate the expression; the result uses the units shown.

    Result=1 \mathrm{Result}=1\ {}

Interpretation. Modely=a+bx.

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Example 18. Residual

Definitions & inputs. Observation5,prediction4.5.

  1. Choose the governing model and isolate the requested quantity.

    e=y−y^e=y-\hat y

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    5−4.55-4.5

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 \mathrm{Result}=0.5\ {}

Interpretation. Signed residual.

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Example 19. Beta-Binomial posterior mean

Definitions & inputs. Prior Beta1,1;7successes3failures.

  1. Choose the governing model and isolate the requested quantity.

    E[p∣D]=(1+7)/(1+1+10)E[p|D]=(1+7)/(1+1+10)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    8/128/12

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6666667 \mathrm{Result}=0.6666667\ {}

Interpretation. Bayesian mean with uniform prior.

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Example 20. Bonferroni threshold

Definitions & inputs. Familyα.05,10tests.

  1. Choose the governing model and isolate the requested quantity.

    αper=α/m\alpha_{per}=\alpha/m

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .05/10.05/10

  3. Evaluate the expression; the result uses the units shown.

    Result=0.005 \mathrm{Result}=0.005\ {}

Interpretation. Union-bound family wise control,not power optimal.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.