MT601M: Probability, Statistics and Theory of Integration Tripura University UG Mathematics – Semester VI
MT601M: Probability, Statistics and Theory of Integration Tripura University UG Mathematics – Semester VI
MT601M: Probability, Statistics and Theory of Integration
Tripura University – UG Course | Semester VI | Paper 7
Complete Study Notes, Concepts, Definitions & Accurate Formula Sheet
📚 Unit 1 – Complete Contents
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UNIT 1: Statistics – Complete Study Notes
1. Measures of Central Tendency
The three most important measures are:
- Arithmetic Mean
- Median
- Mode
Arithmetic Mean
x̄ = Σx / n
x̄ = Σfx / Σf
x̄ = A + (Σfd / Σf)
x̄ = A + h(Σfu / Σf)
Here, A is the assumed mean, d = x − A, u = (x − A)/h, and h is the common class width.
Median
Median = l + [(N/2 − cf) / f]h
Here, l = lower boundary of median class, N = total frequency, cf = cumulative frequency before the median class, f = frequency of median class, and h = class width.
Mode
Here, f₁ is the frequency of the modal class, f₀ is the frequency of the class preceding it, f₂ is the frequency of the class succeeding it.
2. Measures of Dispersion
2.1 Range
R = L − S
L = largest observation and S = smallest observation.
(L − S) / (L + S)
2.2 Quartile Deviation
QD = (Q₃ − Q₁) / 2
(Q₃ − Q₁) / (Q₃ + Q₁)
Quartile deviation is also called the semi-interquartile range.
2.3 Mean Deviation
MD = Σ|x − A| / n
MD = Σf|x − A| / Σf
A may be mean, median or mode. Mean deviation is generally minimum when measured from the median.
Coefficient of MD = MD / A
2.4 Variance
σ² = Σ(x − μ)² / N
σ² = Σf(x − x̄)² / Σf
σ² = Σx²/N − (Σx/N)²
σ² = Σfx²/Σf − (Σfx/Σf)²
2.5 Standard Deviation
σ = √σ²
σ = √[Σ(x − x̄)² / N]
σ = h √[(Σfu²/N) − (Σfu/N)²]
2.6 Coefficient of Variation
CV = (σ / x̄) × 100%
Absolute Measures
- Range
- Quartile Deviation
- Mean Deviation
- Standard Deviation
Relative Measure
- Coefficient of Range
- Coefficient of QD
- Coefficient of MD
- Coefficient of Variation
3. Moments
Raw Moments
Here μ′ᵣ denotes the r-th moment about the origin.
Central Moments
Important central moments:
Relations Between Raw and Central Moments
4. Skewness
Types of Skewness
Positive Skewness
The distribution has a longer tail towards the right.
Mean > Median > Mode
Negative Skewness
The distribution has a longer tail towards the left.
Mean < Median < Mode
Karl Pearson’s Coefficient of Skewness
If mode is not reliable, the empirical relationship Mode ≈ 3 Median − 2 Mean may be used.
Moment Coefficient of Skewness
Relationship:
5. Kurtosis
Coefficient of Kurtosis
Types of Kurtosis
| Type | Condition | Meaning |
|---|---|---|
| Mesokurtic | β₂ = 3 | Normal degree of peakedness |
| Leptokurtic | β₂ > 3 | More peaked than normal |
| Platykurtic | β₂ < 3 | Flatter than normal |
6. Bivariate Frequency Distribution
For example, the marks of students in Mathematics and Physics can be represented using two variables X and Y.
Marginal Frequency Distribution
The marginal distribution of X is obtained by adding all frequencies corresponding to each value or class of X.
Similarly, the marginal distribution of Y is:
Conditional Frequency Distribution
A conditional frequency distribution studies one variable when the other variable is fixed.
7. Pearson’s Correlation Coefficient
r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² Σ(y − ȳ)²]
Short Formula
Properties of Correlation Coefficient
- −1 ≤ r ≤ +1
- r = +1 indicates perfect positive correlation.
- r = −1 indicates perfect negative correlation.
- r = 0 indicates absence of linear correlation.
- Correlation coefficient is unit-free.
- It is unaffected by change of origin and scale.
- r(X,Y) = r(Y,X).
Interpretation
| Value of r | Interpretation |
|---|---|
| +1 | Perfect positive linear correlation |
| Close to +1 | Strong positive correlation |
| 0 | No linear correlation |
| Close to −1 | Strong negative correlation |
| −1 | Perfect negative linear correlation |
8. Regression
Regression Lines
There are two regression lines:
- Regression of Y on X
- Regression of X on Y
Regression Equation of Y on X
Regression Equation of X on Y
Important Properties
- Both regression lines pass through (x̄, ȳ).
- The two regression coefficients have the same sign.
- bxybyx = r².
- Therefore, r = ±√(bxybyx).
- Regression coefficients are independent of change of origin.
- Regression coefficients are affected by change of scale.
Least Squares Method
The method of least squares determines the line that minimizes the sum of squared vertical deviations between observed and fitted values.
Solving these two simultaneous equations gives a and b.
9. Curve Fitting
9.1 Fitting a Second-Degree Polynomial
The general equation is:
Using the method of least squares, minimize:
The normal equations are:
These three simultaneous equations are solved to obtain the constants a, b and c.
9.2 Exponential Curve
The exponential model is:
Taking logarithms:
Put:
Then:
This is a straight-line equation and can be fitted using the least-squares method.
After finding A and B:
10. Unit 1 Master Formula Sheet
⚡ Quick Revision Points
Central Tendency – What should I remember?
Dispersion – Most important formulas
QD = (Q₃ − Q₁)/2;
σ = √σ²;
CV = (σ/x̄) × 100.
Correlation – Most important facts
Regression – Most important facts
Curve Fitting – Exam approach
🎯 MT601M Unit 1 Examination Checklist
- Definitions of all measures of dispersion
- Numerical problems on Range and Quartile Deviation
- Mean Deviation calculations
- Variance and Standard Deviation
- Coefficient of Variation comparison problems
- Raw and central moments
- Skewness using Pearson and moment methods
- Kurtosis and β₂, γ₂
- Bivariate frequency distributions
- Marginal and conditional distributions
- Pearson’s correlation coefficient
- Properties of correlation coefficient
- Regression coefficients and regression equations
- Least-squares straight-line fitting
- Second-degree polynomial fitting
- Exponential curve fitting using logarithms
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MT601M: Probability, Statistics and Theory of Integration Tripura University UG Mathematics – Semester VI
