IB Mathematics AA Variance of random variable Study Notes
IB Mathematics AA Variance of random variable Study Notes
IB Mathematics AA Variance of random variable Notes Offer a clear explanation of Use of Variance of random variable, including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Variance of random variable.
Variance of a Random Variable
Introduction
The variance of a random variable measures the spread or dispersion of its values around the expected value (mean). It is a fundamental concept in probability theory and statistics, used to quantify uncertainty in various fields such as economics, physics, engineering, and finance.
Key Concepts
1. Variance of a Discrete Random Variable
- The variance of a discrete random variable \( X \) is given by:
\( \text{Var}(X) = E(X^2) – [E(X)]^2 \)
- \( E(X) \) = Expected value (mean) of \( X \).
- \( E(X^2) \) = Expected value of \( X^2 \), calculated as:
\( E(X^2) = \sum x^2 P(X=x) \)
- \( \text{Var}(X) \) measures how much \( X \) deviates from its mean.
2. Variance of a Continuous Random Variable
- For a continuous random variable \( X \) with probability density function (PDF) \( f(x) \), variance is calculated as:
\( \text{Var}(X) = \int_{-\infty}^{\infty} (x – \mu)^2 f(x) dx \)
- \( \mu = E(X) = \int_{-\infty}^{\infty} x f(x) dx \)
3. Notation and Important Formulas
- Expected Value: \( E(X) \) represents the mean of \( X \).
- Second Moment: \( E(X^2) \) is used to compute variance.
- Standard Deviation: \( \sigma(X) = \sqrt{\text{Var}(X)} \).
Guidance, Clarification, and Syllabus Links
- Understanding variance requires knowledge of discrete and continuous random variables.
- Links to discrete random variables (SL 4.7).
- Connections to probability density functions (PDFs).
4. Mode and Median of Continuous Random Variables
- Mode: The value \( x \) at which \( f(x) \) has a maximum.
- Median: The value \( m \) satisfying:
\( \int_{-\infty}^{m} f(x) dx = 0.5 \)
5. The Effect of Linear Transformations
- For a transformed variable \( Y = aX + b \):
\( E(aX + b) = aE(X) + b \)
\( \text{Var}(aX + b) = a^2 \text{Var}(X) \)
- Used in financial models and risk assessment.
Connections and Applications
1. Other Contexts
- Insurance: Expected value and variance help calculate risk premiums.
- Economics & Business: Used for financial forecasting and decision-making.
- Poisson Distribution: Often used in modeling rare events.
2. Theory of Knowledge (TOK) Considerations
- Is mathematics more or less useful than other fields for solving real-world problems?
- Does variance always provide a meaningful measure of spread?
3. Enrichment: Relationship Between Interquartile Range and Standard Deviation
- For normally distributed data, interquartile range (IQR) and standard deviation (\( \sigma \)) are related by:
\( \text{IQR} \approx 1.35\sigma \)
Solved Example
Problem: A discrete random variable \( X \) has the following probability distribution:
X | -1 | 0 | 2 |
---|---|---|---|
P(X) | 0.2 | 0.5 | 0.3 |
Step 1: Compute \( E(X) \)
\( E(X) = (-1)(0.2) + (0)(0.5) + (2)(0.3) \)
\( = -0.2 + 0 + 0.6 = 0.4 \)
Step 2: Compute \( E(X^2) \)
\( E(X^2) = (-1)^2(0.2) + (0)^2(0.5) + (2)^2(0.3) \)
\( = 1(0.2) + 0 + 4(0.3) = 0.2 + 1.2 = 1.4 \)
Step 3: Compute \( \text{Var}(X) \)
\( \text{Var}(X) = E(X^2) – [E(X)]^2 \)
\( = 1.4 – (0.4)^2 = 1.4 – 0.16 = 1.24 \)
Step 4: Compute Standard Deviation
\( \sigma(X) = \sqrt{1.24} \approx 1.11 \)
Interpretation: The variance is 1.24, and the standard deviation is 1.11, indicating how much \( X \) deviates from its mean.
IB Mathematics AA SL Variance of random variable Exam Style Worked Out Questions
Question
A biased coin is tossed five times. The probability of obtaining a head in any one throw is \(p\).
Let \(X\) be the number of heads obtained.
Find, in terms of \(p\), an expression for \({\text{P}}(X = 4)\).
(i) Determine the value of \(p\) for which \({\text{P}}(X = 4)\) is a maximum.
(ii) For this value of \(p\), determine the expected number of heads.
▶️Answer/Explanation
Markscheme
\(X \sim {\text{B}}(5,{\text{ }}p)\) (M1)
\({\text{P}}(X = 4) = \left( {\begin{array}{*{20}{c}} 5 \\ 4 \end{array}} \right){p^4}(1 – p)\) (or equivalent) A1
[2 marks]
(i) \(\frac{{\text{d}}}{{{\text{d}}p}}(5{p^4} – 5{p^5}) = 20{p^3} – 25{p^4}\) M1A1
\(5{p^3}(4 – 5p) = 0 \Rightarrow p = \frac{4}{5}\) M1A1
Note: Do not award the final A1 if \(p = 0\) is included in the answer.
(ii) \({\text{E}}(X) = np = 5\left( {\frac{4}{5}} \right)\) (M1)
\( = 4\) A1
[6 marks]
Examiners report
This question was generally very well done and posed few problems except for the weakest candidates.
This question was generally very well done and posed few problems except for the weakest candidates.
Question
A biased coin is tossed five times. The probability of obtaining a head in any one throw is \(p\).
Let \(X\) be the number of heads obtained.
Find, in terms of \(p\), an expression for \({\text{P}}(X = 4)\).
(i) Determine the value of \(p\) for which \({\text{P}}(X = 4)\) is a maximum.
(ii) For this value of \(p\), determine the expected number of heads.
▶️Answer/Explanation
Markscheme
\(X \sim {\text{B}}(5,{\text{ }}p)\) (M1)
\({\text{P}}(X = 4) = \left( {\begin{array}{*{20}{c}} 5 \\ 4 \end{array}} \right){p^4}(1 – p)\) (or equivalent) A1
[2 marks]
(i) \(\frac{{\text{d}}}{{{\text{d}}p}}(5{p^4} – 5{p^5}) = 20{p^3} – 25{p^4}\) M1A1
\(5{p^3}(4 – 5p) = 0 \Rightarrow p = \frac{4}{5}\) M1A1
Note: Do not award the final A1 if \(p = 0\) is included in the answer.
(ii) \({\text{E}}(X) = np = 5\left( {\frac{4}{5}} \right)\) (M1)
\( = 4\) A1
[6 marks]
Examiners report
This question was generally very well done and posed few problems except for the weakest candidates.
This question was generally very well done and posed few problems except for the weakest candidates.