Calculating Mean, Variance, and Percentiles

Introduction

Key Concepts

1. Mean (Expected Value)

The mean, often referred to as the expected value ($E[X]$), is a measure of the central tendency of a continuous random variable. It represents the long-run average outcome of a random variable over numerous trials.

Mathematically, the mean of a continuous random variable $X$ with probability density function (PDF) $f_X(x)$ is calculated as: $$ E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) \, dx $$

**Example:** Consider a continuous random variable $X$ with PDF $f_X(x) = 2x$ for $0 \leq x \leq 1$. To find the mean: $$ E[X] = \int_{0}^{1} x \cdot 2x \, dx = 2 \int_{0}^{1} x^2 \, dx = 2 \left[ \frac{x^3}{3} \right]_0^1 = 2 \cdot \frac{1}{3} = \frac{2}{3} $$

2. Variance

Variance measures the dispersion of a continuous random variable around its mean. It quantifies the degree to which each data point differs from the mean of the distribution.

The variance ($Var(X)$) is defined as: $$ Var(X) = E[(X - \mu)^2] = \int_{-\infty}^{\infty} (x - \mu)^2 \cdot f_X(x) \, dx $$ where $\mu = E[X]$.

Alternatively, variance can be computed using the formula: $$ Var(X) = E[X^2] - (E[X])^2 $$ where: $$ E[X^2] = \int_{-\infty}^{\infty} x^2 \cdot f_X(x) \, dx $$

**Example:** Using the previous PDF $f_X(x) = 2x$ for $0 \leq x \leq 1$, first calculate $E[X^2]$: $$ E[X^2] = \int_{0}^{1} x^2 \cdot 2x \, dx = 2 \int_{0}^{1} x^3 \, dx = 2 \left[ \frac{x^4}{4} \right]_0^1 = 2 \cdot \frac{1}{4} = \frac{1}{2} $$ Then, compute the variance: $$ Var(X) = \frac{1}{2} - \left(\frac{2}{3}\right)^2 = \frac{1}{2} - \frac{4}{9} = \frac{9}{18} - \frac{8}{18} = \frac{1}{18} $$

3. Percentiles

Percentiles indicate the relative standing of a particular value within a dataset. The $p^{th}$ percentile ($P_p$) is the value below which $p\%$ of the data falls.

For a continuous random variable $X$ with Cumulative Distribution Function (CDF) $F_X(x)$, the $p^{th}$ percentile is found by solving: $$ F_X(P_p) = p $$ where $F_X(x) = \int_{-\infty}^x f_X(t) \, dt$.

**Example:** Using $f_X(x) = 2x$ for $0 \leq x \leq 1$, find the 75th percentile ($P_{75}$): First, determine the CDF: $$ F_X(x) = \int_{0}^{x} 2t \, dt = [t^2]_0^x = x^2 $$ Set $F_X(P_{75}) = 0.75$: $$ P_{75}^2 = 0.75 \\ P_{75} = \sqrt{0.75} \approx 0.866 $$ So, the 75th percentile is approximately 0.866.

4. Probability Density Function (PDF) and Cumulative Distribution Function (CDF)

The PDF, $f_X(x)$, describes the likelihood of a continuous random variable $X$ taking on a specific value. The CDF, $F_X(x)$, gives the probability that $X$ will be less than or equal to $x$.

For calculations involving mean, variance, and percentiles, both PDF and CDF are essential tools. The PDF is used in integrating to find expected values, while the CDF is directly used in determining percentiles.

5. Skewness and Kurtosis

While not explicitly required in the key concepts, understanding skewness and kurtosis provides deeper insights into the distribution's shape. Skewness measures the asymmetry, and kurtosis measures the "tailedness" of the distribution. These concepts are useful in advanced statistical analyses but are beyond the scope of basic mean, variance, and percentile calculations.

6. Applications in AS & A Level Mathematics

Calculating mean, variance, and percentiles is integral to numerous mathematical applications, including hypothesis testing, confidence interval estimation, and regression analysis. Students engage with these concepts to interpret data, assess variability, and make informed predictions based on probability distributions.

Advanced Concepts

1. Moment Generating Functions (MGFs)

Moment Generating Functions are powerful tools that simplify the computation of moments (mean, variance, etc.) of a random variable. For a continuous random variable $X$, the MGF, $M_X(t)$, is defined as: $$ M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty} e^{tX} f_X(x) \, dx $$

The $n^{th}$ moment of $X$ can be obtained by taking the $n^{th}$ derivative of $M_X(t)$ evaluated at $t=0$: $$ E[X^n] = M_X^{(n)}(0) $$

**Example:** Using the earlier PDF $f_X(x) = 2x$ for $0 \leq x \leq 1$, find the MGF: $$ M_X(t) = \int_{0}^{1} e^{tx} \cdot 2x \, dx $$ This integral may not have a closed-form solution but can be evaluated using integration by parts or series expansion techniques for specific values of $t$.

2. Covariance and Correlation

Extending beyond single random variables, covariance measures the joint variability of two random variables, while correlation standardizes this measure. For two continuous random variables $X$ and $Y$ with joint PDF $f_{X,Y}(x,y)$: $$ Cov(X,Y) = E[(X - E[X])(Y - E[Y])] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} (x - E[X])(y - E[Y]) f_{X,Y}(x,y) \, dx \, dy $$ $$ \rho_{X,Y} = \frac{Cov(X,Y)}{\sqrt{Var(X) Var(Y)}} $$

These measures are pivotal in multivariate statistics, allowing assessments of relationships between variables.

3. Central Limit Theorem (CLT)

The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size becomes large, regardless of the original distribution's shape, provided the variance is finite.

Mathematically, if $X_1, X_2, ..., X_n$ are independent and identically distributed (i.i.d.) random variables with mean $\mu$ and variance $\sigma^2$, then: $$ \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0,1) \quad \text{as} \quad n \to \infty $$ where $\bar{X}$ is the sample mean.

**Implications:** The CLT justifies the use of normal distribution approximations in various statistical procedures, particularly in hypothesis testing and confidence interval construction.

4. Transformations of Random Variables

Transforming random variables involves deriving the distribution of a new variable defined as a function of an existing variable. For instance, if $Y = g(X)$, finding $f_Y(y)$ involves: $$ f_Y(y) = f_X(x) \left| \frac{dx}{dy} \right| $$ where $x = g^{-1}(y)$.

**Example:** Let $Y = X^2$ where $X$ has PDF $f_X(x) = 1$ for $0 \leq x \leq 1$. $$ f_Y(y) = f_X(\sqrt{y}) \cdot \frac{1}{2\sqrt{y}} = \frac{1}{2\sqrt{y}} \quad \text{for} \quad 0 \leq y \leq 1 $$

5. Multivariate Distributions

While single-variable distributions deal with one random variable, multivariate distributions handle multiple random variables simultaneously. Key concepts include joint PDFs, marginal distributions, and conditional distributions.

**Example:** For two continuous random variables $X$ and $Y$, the joint PDF $f_{X,Y}(x,y)$ describes the probability distribution over the two-dimensional space. Marginal PDFs are obtained by integrating the joint PDF over the other variable: $$ f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) \, dy $$ $$ f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) \, dx $$

6. Bayesian Inference Connections

Bayesian inference integrates prior knowledge with observed data to update the probability estimates for a hypothesis. Calculating mean, variance, and percentiles are integral in determining posterior distributions, which are central to Bayesian analysis.

**Example:** In Bayesian statistics, the posterior mean serves as an updated estimate incorporating both prior beliefs and new evidence, often calculated using integrals similar to those for expected values in continuous random variables.

7. Advanced Applications in Other Disciplines

The concepts of mean, variance, and percentiles extend beyond pure mathematics into various fields:

• Economics: Used in risk assessment and investment portfolio optimization.
• Engineering: Applied in quality control and reliability testing.
• Medicine: Essential in biostatistics for analyzing clinical trial data.
• Machine Learning: Fundamental in algorithm performance evaluation and data preprocessing.

Understanding these advanced connections broadens the applicability and relevance of statistical measures in real-world scenarios.

8. Numerical Methods for Complex Integrals

In cases where analytical solutions for mean, variance, or percentiles are infeasible, numerical integration techniques such as the Trapezoidal Rule or Simpson's Rule are employed to approximate the required integrals.

**Example:** For a PDF where $f_X(x) = e^{-x}$ for $x \geq 0$, calculating $E[X]$: $$ E[X] = \int_{0}^{\infty} x e^{-x} \, dx $$ While this integral has a closed-form solution, more complex PDFs might require numerical methods for evaluation.

Comparison Table

Summary and Key Takeaways