medium3 marks

Probability and Statistics

Random variables, probability distributions (binomial, Poisson, normal, exponential), conditional probability, Bayes theorem, mean, median, mode, standard deviation.

12 Topics

20h prep

23.08% subject weight

12 Topics

Random variables

Variable whose value depends on random outcomes.

1m2/10

📌 Key FormulaE[X] = sum xP(x) (discrete) or ∫ x f(x) dx (continuous).

Conditional probability

Probability of an event given another event has occurred.

1m2/10

📌 Key FormulaP(A|B) = P(A∩B)/P(B), P(B)>0.

Binomial distributions

Discrete distribution for number of successes in n independent trials.

2/10

📌 Key FormulaP(X=k)= C(n,k) p^k (1-p)^{n-k}, mean = np, variance = np(1-p).

Uniform distribution

All outcomes equally likely over an interval.

1/10

📌 Key FormulaPDF f(x)=1/(b-a) for x∈[a,b]; mean = (a+b)/2, variance = (b-a)^2/12.

Normal distributions

Bell-shaped continuous distribution, symmetric about mean.

1m2/10

📌 Key FormulaPDF: (1/(σ√(2π))) e^{-(x-μ)^2/(2σ^2)}. Standard normal: Z = (X-μ)/σ.

Exponential distributions

Describes time between Poisson events.

2/10

📌 Key FormulaPDF: λe^{-λx} for x≥0; mean=1/λ, variance=1/λ^2.

Poisson distributions

Discrete distribution for number of events in fixed interval.

2/10

📌 Key FormulaP(X=k)= e^{-λ} λ^k/k!, mean=variance=λ.

Mean deviation

Average absolute deviation from mean.

1/10

📌 Key FormulaMean deviation = (1/n) Σ|x_i - mean|.

Median deviation

Average absolute deviation from median.

1/10

📌 Key FormulaMedian deviation = (1/n) Σ|x_i - median|.

Mode deviation

Deviation measure based on mode (rarely used alone).

1/10

📌 Key FormulaNot standard; sometimes defined as |x - mode|.

Standard deviation

Square root of variance; measure of dispersion.

1m2/10

📌 Key Formulaσ = sqrt( (1/n) Σ(x_i - μ)^2 ).

Bayes theorem

Calculates conditional probability from inverse probabilities.

1m3/10

📌 Key FormulaP(A|B) = P(B|A)P(A)/P(B).

medium3 marks

Probability and Statistics

Random variables, probability distributions (binomial, Poisson, normal, exponential), conditional probability, Bayes theorem, mean, median, mode, standard deviation.

12 Topics

20h prep

23.08% subject weight

12 Topics

Random variables

Variable whose value depends on random outcomes.

1m2/10

📌 Key FormulaE[X] = sum xP(x) (discrete) or ∫ x f(x) dx (continuous).

Conditional probability

Probability of an event given another event has occurred.

1m2/10

📌 Key FormulaP(A|B) = P(A∩B)/P(B), P(B)>0.

Binomial distributions

Discrete distribution for number of successes in n independent trials.

2/10

📌 Key FormulaP(X=k)= C(n,k) p^k (1-p)^{n-k}, mean = np, variance = np(1-p).

Uniform distribution

All outcomes equally likely over an interval.

1/10

📌 Key FormulaPDF f(x)=1/(b-a) for x∈[a,b]; mean = (a+b)/2, variance = (b-a)^2/12.

Normal distributions

Bell-shaped continuous distribution, symmetric about mean.

1m2/10

📌 Key FormulaPDF: (1/(σ√(2π))) e^{-(x-μ)^2/(2σ^2)}. Standard normal: Z = (X-μ)/σ.

Exponential distributions

Describes time between Poisson events.

2/10

📌 Key FormulaPDF: λe^{-λx} for x≥0; mean=1/λ, variance=1/λ^2.

Poisson distributions

Discrete distribution for number of events in fixed interval.

2/10

📌 Key FormulaP(X=k)= e^{-λ} λ^k/k!, mean=variance=λ.

Mean deviation

Average absolute deviation from mean.

1/10

📌 Key FormulaMean deviation = (1/n) Σ|x_i - mean|.

Median deviation

Average absolute deviation from median.

1/10

📌 Key FormulaMedian deviation = (1/n) Σ|x_i - median|.

Mode deviation

Deviation measure based on mode (rarely used alone).

1/10

📌 Key FormulaNot standard; sometimes defined as |x - mode|.

Standard deviation

Square root of variance; measure of dispersion.

1m2/10

📌 Key Formulaσ = sqrt( (1/n) Σ(x_i - μ)^2 ).

Bayes theorem

Calculates conditional probability from inverse probabilities.

1m3/10

📌 Key FormulaP(A|B) = P(B|A)P(A)/P(B).