Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Probability and Statistics → Probability Distributions ↓


Normal Distribution


When we talk about probability and statistics, one of the most important topics we come across is the normal distribution. It is like the cornerstone for various statistical concepts and theories. Just imagine, you are looking at the height of students in a class. Most of the students will be around the average height, while very few students will be very tall or very short. This kind of pattern is common in many natural phenomena and we call it normal distribution.

What is normal distribution?

The normal distribution is a type of continuous probability distribution for a real-valued random variable. It is sometimes informally called a "bell curve" because of its distinctive shape (resembling a bell). The normal distribution is symmetric about its mean, indicating that data near the mean occur more often than data far from the mean.

Characteristics of a Normal Distribution:
1. Symmetrical: The left side is a mirror image of the right side.
2. Bell-shaped: The shape of the distribution is often referred to as bell-shaped.
3. Mean, Median, and Mode are equal: In a perfectly normal distribution, these values are all the same and occur at the highest peak.
4. Defined by two parameters: The mean (average) and the standard deviation (measure of spread).
Meaning

Importance of normal distribution

The normal distribution is extremely important in statistics and is often used in the natural and social sciences to represent real-valued random variables whose distribution is not known. This is because:

  • This provides the basis for the theory of statistical inference.
  • It accurately describes many phenomena of the natural world.
  • The properties of the normal distribution make it important for calculations in statistical studies.

Mathematical representation

The normal distribution can be represented mathematically using its probability density function (PDF). The PDF is a function that describes the probability of a random variable taking on a particular value.

The formula for the probability density function of a normal distribution is:
f(x | μ, σ) = (1 / (σ * sqrt(2π))) * exp(-(x - μ)² / (2σ²))
Where:
- μ is the mean or expectation.
- σ is the standard deviation.
- π is a constant (approximately 3.14159).
- exp is the exponential function.

Understanding the bell curve

The bell curve is one of the reasons why the normal distribution is famous. Let's see what the curve represents. The highest point on the curve represents the mean, median, and mode, which are exactly the same in a normal distribution.

Standard deviations determine the spread of data points. The higher the standard deviation, the wider the bell curve will appear. A smaller standard deviation will result in a steeper bell curve.

Meaning -1σ +1σ

Applications of normal distribution

The normal distribution is not just an abstract concept, it has practical applications in various fields:

  • In finance: It is used in the Black–Scholes option pricing model, and to assess the probability of positive or negative performance of stocks and portfolios.
  • In biology: It is used to model biological data, such as height and blood pressure readings.
  • In quality control: Companies use the normal distribution to understand variation in their manufacturing processes.

Examples and practice problems

Let us work out some examples to understand the concept of normal distribution better.

Example 1:

Suppose the weights of apples in an orchard follow a normal distribution with a mean of 150 g and a standard deviation of 15 g. What is the probability that a randomly selected apple weighs between 135 g and 165 g?

Here we need to calculate the probability that the weight of an apple x is between 135 and 165 grams.
P(135 < x < 165)
Using standard normal distribution tables or a calculator:
z-score = (X - μ) / σ
z(135) = (135 - 150) / 15 = -1
z(165) = (165 - 150) / 15 = 1
Therefore, P(-1 < z < 1) = 0.6826 or 68.26%
This implies that there is a 68.26% chance that the weight of a randomly chosen apple is between 135 and 165 grams.

Example 2:

A teacher found that the marks of her students on a test are normally distributed with a mean of 70 and a standard deviation of 10. What is the percentage of students who scored more than 85 marks?

Here, we need to find the probability that a student's score x is greater than 85.
P(x > 85)
z = (X - μ) / σ
z(85) = (85 - 70) / 10 = 1.5
Using standard normal distribution tables:
P(z < 1.5) = 0.9332
Therefore, P(x > 85) = 1 - 0.9332 = 0.0668 or 6.68%
Thus, 6.68% of students scored above 85.

Conclusion

The normal distribution is a fundamental concept in statistics, important for theoretical as well as applied statistical analysis. Its properties of symmetry, defined mean and standard deviation allow it to effectively model a variety of real-world phenomena. Whether in exams, manufacturing, biology or finance, understanding the normal distribution helps to interpret data and make informed decisions. With its wide applications and ease of use, mastering the normal distribution equips you with a powerful tool for analyzing real-world data.


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Binomial Distribution
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Normal Distribution

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