MTH1005 Probability and Statistics (Cheat Sheet)

MTH1005 Probability and Statistics (Cheat Sheet) §

MTH1005 Probability and Statistics Cheat Sheet by William Fayers :)

Basic Probability Concepts §

• Multiplicative Rule of Probability:
  • Independent Events: $P(A\cap B)=P(A)\times P(B)$.
  • Dependent Events: $P(A\cap B)=P(A)\times P(B|A)$.
• Conditional Probability: $P(B|A)=\frac{P(A\cap B)}{P(A)}$.
• Total Probability Law: $P(B)={\sum }_{i=1}^{n}P(B|A_i)\times P(A_i)$.
• Bayes’ Formula: $P(A|B)=\frac{P(B|A)\times P(A)}{P(B)}$.

Useful empty Venn diagram for probability visualisation (to understand questions & their solutions):

Random Variables §

• Discrete Random Variables: Ensure probabilities of all outcomes sum to 1.
• Probability Mass Functions (PMF): $P(X=x)$.
• Cumulative Distribution Functions (CDF): $F(x)=P(X\le x)$.
• Continuous Random Variables: Probabilities over intervals, not points.
• Probability Density Functions (PDF): $P(a\le X\le b)={\int }_a^{b}f(x)dx$.
• CDF for Continuous Variables: $F(x)={\int }_{-\infty }^{x}f(t)dt$.

Statistical Measures §

• Expectation and Variance:
  • Expectation Value: $E(X)=\sum x_{i}P(X=x_i)$ or $E(X)=\int xf(x)dx$.
  • Variance: $\text{Var}(X)=E[(X-\mu {)}^2]$ or $\text{Var}(X)=E(X^2)-E(X{)}^2$.
  • Mappings: $E(aX+b)=aE(X)+b$, $\text{Var}(aX+b)=a^2\text{Var}(X)$.
• Joint Random Variables:
  • Joint PMF: $P(X=x,Y=y)$.
  • Joint PDF: $f_{X,Y}(x,y)$.
  • Marginal Distributions: Derived from joint distribution by integrating out other variables.
• Covariance and Correlations:
  • Covariance: $\text{Cov}(X,Y)=E[(X-E(X))(Y-E(Y))]$.
  • Correlation: ${\rho }_{X,Y}=\frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}}$.

Data Analysis §

• Median and Quantiles:
  • Median: Middle value of sorted data.
  • Quantiles (Quartiles and Percentiles): Divide data into intervals with equal probabilities.
• Interquartile Distance (IQR): $\text{IQR}=Q3-Q1$.
• Inequalities:
  • Markov’s and Chebyshev’s Inequalities:
    • Markov: $P(X\ge a)\le \frac{E(X)}{a}$.
    • Chebyshev: $P(|X-\mu |\ge k\sigma )\le \frac{1}{k^2}$.

Common Distributions §

1. Uniform Distribution §

• Expectation (Mean): $E(X)=\frac{a+b}{2}$
• Variance: $\text{Var}(X)=\frac{(b-a{)}^2}{12}$
• Probability Mass Function (PMF): $P(X=x)=\frac{1}{b-a+1}\text{for}x=a,a+1,\dots ,b$ Every outcome within the range from $a$ to $b$ is equally likely.

2. Geometric Distribution §

• Expectation (Mean): $E(X)=\frac{1}{p}$
• Variance: $\text{Var}(X)=\frac{1-p}{p^2}$
• Probability Mass Function (PMF): $P(X=k)=(1-p{)}^{k-1}p\text{for}k=1,2,3,\dots$ Represents the probability of getting the first success on the $k$-th trial.

3. Exponential Distribution §

• Expectation (Mean): $E(X)=\frac{1}{\lambda }$
• Variance: $\text{Var}(X)=\frac{1}{{\lambda }^2}$
• Probability Density Function (PDF): $f(x)=\lambda e^{-\lambda x}\text{for}x\ge 0$ Describes the time between events in a Poisson point process.

4. Normal Distribution §

• Expectation (Mean): $E(X)=\mu$
• Variance: $\text{Var}(X)={\sigma }^2$
• Probability Density Function (PDF): $f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}$ Common for continuous data clustering around a mean.

5. Poisson Distribution §

• Expectation (Mean): $E(X)=\lambda$
• Variance: $\text{Var}(X)=\lambda$
• Probability Mass Function (PMF): $P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!}\text{for}k=0,1,2,\dots$ Suitable for modelling the frequency of events within a fixed interval.

6. Binomial Distribution §

• Expectation (Mean): $E(X)=np$
• Variance: $\text{Var}(X)=np(1-p)$
• Probability Mass Function (PMF): $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}\text{for}k=0,1,2,\dots ,n$ Describes the number of successes in a series of independent Bernoulli trials.

Decision Guide for Distributions §

1. Outcomes type:
   • Discrete: Focuses on distinct, separate outcomes. Go to Step 2.
   • Continuous: Concerns measurable quantities that can vary. Go to Step 4.
2. Trial Independence:
   • Independent with two outcomes: Each outcome doesn’t affect the others, with success/failure options. Go to Step 3.
   • Not independent or different outcomes: Outcomes affect each other or there are multiple types of outcomes. Go to Step 5.
3. Number of trials and interest in successes:
   • Known trials and interest in successes: Binomial Distribution, used when the number of trials and the desire to determine the number of specific outcomes (like successes) is known.
   • Interest in number of trials for first success: Geometric Distribution, applies when focusing on the count of trials needed to achieve a first success.
4. Time or space between events:
   • Events over time/space: Exponential Distribution, ideal for modelling time between continuous, independent events.
   • Data clustering around mean: Normal Distribution, effective when large data sets naturally cluster around a central value.
5. Probability equality across range:
   • Equal across range: Uniform Distribution, where each outcome in a range is equally likely.
   • Not equal across range: Poisson Distribution, suited for counting the occurrences of events over a fixed interval, where events happen with a known but uneven rate.

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