Permutation is the arrangement of objects in a specific order. Combination is the selection of objects without considering the order.
P(n, r) = n! / (n - r)!
C(n, r) = n! / [r! (n - r)!]
How many ways can you arrange 3 letters out of 5?
Permutation: P(5,3) = 5! / (5-3)! = 60
Combination: C(5,3) = 5! / (3!2!) = 10
Set = {A, B, C, D, E}
Permutation (3 items): ABC, ACB, BAC, BCA, CAB, CBA...
Combination (3 items): ABC, ABD, ABE, ACD...
Repetition allows objects to be selected more than once. Constrained repetition limits the number of times an object can be selected.
Total = n^r
(Choose r items from n, allowing repetition)
How many 2-digit numbers using 3 digits {1,2,3} with repetition?
3^2 = 9 combinations: 11, 12, 13, 21, 22, ...
Binomial coefficients are the values of C(n, r), used in the Binomial Theorem and Pascal's Triangle.
C(n, r) = n! / [r!(n-r)!]
Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
The Binomial Theorem gives a way to expand expressions of the form (a + b)n.
(a + b)^n = Σ [C(n, k) * a^(n-k) * b^k] for k = 0 to n
(a + b)^2 = C(2,0)a^2 + C(2,1)ab + C(2,2)b^2
= a^2 + 2ab + b^2
A frequency distribution organizes raw data into a table showing the frequency (number of times) each value appears.
Example:
Data: 2, 3, 2, 5, 3, 2, 4, 5
Frequency Table:
Value | Frequency
------|----------
2 | 3
3 | 2
4 | 1
5 | 2
A histogram is a graphical representation of a frequency distribution using bars.
A frequency polygon is a line graph created by joining midpoints of class intervals.
Textual Histogram:
[10-20]: ||||
[20-30]: ||||||
[30-40]: |||
[40-50]: |||||
Average of the data values.
Mean = (2 + 4 + 6 + 8) / 4 = 5
The middle value when data is arranged in order.
Data: 3, 5, 7 —> Median = 5
If even count: average of middle two numbers.
The value that appears most frequently.
Data: 2, 3, 3, 4, 5 —> Mode = 3
The average of absolute deviations from the mean.
Mean = 4, Data = 3, 4, 5
Deviations = 1, 0, 1 —> Mean Deviation = (1+0+1)/3 = 0.67
Shows how much data deviates from the mean. Formula:
σ = √[Σ(x - mean)² / N]
Example: Data = 2, 4, 4, 4, 5, 5, 7, 9
Mean = 5, σ ≈ 2
Moments are used to describe the shape characteristics of a distribution.
Measures asymmetry of data:
Diagram (Text):
Left-skewed: * * * * * * * *
Symmetrical: * * * * * * *
Right-skewed: * * * * * * * *
Measures the peakedness of the distribution.
Diagram (Text):
Leptokurtic: ^
Mesokurtic: ^
Platykurtic: ^
Probability: The measure of the likelihood that an event will occur.
If 'S' is the sample space and 'E' is an event, then:
P(E) = Number of favorable outcomes / Total number of outcomes
The probability of an event A given that event B has occurred.
P(A | B) = P(A ∩ B) / P(B), provided P(B) ≠ 0
A probability distribution assigns probabilities to all possible outcomes of a random variable.
Example:
X: 0 1 2
P(X): 0.2 0.5 0.3
The expected value (mean) of a random variable is the weighted average of all possible values.
E(X) = Σ [x * P(x)]
Example:
X: 1 2 3
P(X): 0.2 0.5 0.3
E(X) = 1*0.2 + 2*0.5 + 3*0.3 = 2.1
Used when there are fixed number of trials (n), each with two outcomes: success or failure.
Formula: P(X = r) = nCr * p^r * (1-p)^(n-r)
Example: n = 4, p = 0.5
P(X = 2) = 4C2 * (0.5)^2 * (0.5)^2 = 0.375
Used for rare events occurring over fixed intervals of time or space.
Formula: P(X = r) = (λ^r * e^(-λ)) / r!
Example: λ = 3, r = 2
P(X=2) = (3^2 * e^-3) / 2! ≈ 0.224
A continuous distribution that is symmetric and bell-shaped. Mean = Median = Mode.
Formula (standard normal): Z = (X - μ) / σ
Example: μ = 100, σ = 15, X = 115
Z = (115 - 100) / 15 = 1.0
Approximations:
Binomial(n, p) ≈ Poisson(λ = np) if n→∞, p→0
Binomial(n, p) ≈ Normal(μ = np, σ = √npq) if n is large
Poisson(λ) ≈ Normal(μ = λ, σ = √λ) if λ is large
Correlation measures the degree of relationship between two variables.
A linear relationship exists when the change in one variable causes a proportional change in another.
The most common measure is Pearson’s correlation coefficient (r):
r = Σ[(X - X̄)(Y - Ȳ)] / √[Σ(X - X̄)² * Σ(Y - Ȳ)²]
Value of r lies between -1 and 1.
Regression is used to predict the value of one variable based on another.
The regression line minimizes the sum of squares of the vertical distances from data points to the line.
Curve fitting is the process of finding a curve that best fits the given data using the method of least squares.
This method minimizes the sum of the squares of the errors (deviations) between the observed and fitted values.
A straight line fit: Y = a + bX, where a and b are calculated using least squares method.
A curve of the form Y = a + bX + cX² fitted to data using least squares approach.
The Chi-square (χ²) test is a statistical test to determine if there is a significant difference between observed and expected frequencies.
χ² = Σ[(O - E)² / E]
Where O = observed frequency, E = expected frequency.
Used to test the independence of two categorical variables.
Used for testing the independence between row and column variables in a contingency table.
A measure derived from Chi-square to express the degree of association between two attributes.
C = √[χ² / (χ² + N)], where N = total number of observations.
The sample mean is the average of a set of sample data. It is used to estimate the population mean. The sample variance measures how much the data points differ from the sample mean.
Sample Mean (x̄) = Σx / n
Sample Variance (s²) = Σ(x - x̄)² / (n - 1)
The t-test is used to determine if there is a significant difference between the means of two groups, particularly when sample sizes are small and population variance is unknown.
t = (x̄ - μ) / (s / √n)
Where:
x̄ = sample mean, μ = population mean, s = sample standard deviation, n = sample size
Hypothesis testing is used to test an assumption regarding a population parameter. A hypothesis can be either null (H₀) or alternative (H₁). The goal is to determine whether the evidence is strong enough to reject the null hypothesis.
Null Hypothesis (H₀): Assumes no effect or difference.
Alternative Hypothesis (H₁): Assumes there is an effect or difference.
p-value: Probability of observing data at least as extreme as the current data under the null hypothesis.
The degree of freedom refers to the number of independent values or quantities which can be assigned to a statistical distribution. It is used in the calculation of the t-distribution and chi-square tests.
Degree of freedom (df) = n - 1
Where n is the sample size.
The Z-test is used to test the hypothesis about the population mean when the population variance is known or the sample size is large (n > 30). It compares the observed sample mean with the population mean.
Z = (x̄ - μ) / (σ / √n)
Where:
x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size
Small sampling refers to samples with fewer than 30 data points. Large sampling refers to samples with more than 30 data points. Statistical tests like t-tests and Z-tests are applied differently based on the sample size.
The Monte Carlo method is a statistical technique that uses random sampling to obtain numerical results. It is commonly used in simulations, optimization problems, and modeling complex systems.
Monte Carlo Simulation Steps:
1. Define the problem and random variables.
2. Generate random numbers for variables.
3. Run simulations and collect results.
4. Analyze the results to estimate the solution.