# [[Finding the Mean of a Normal Distribution]] (Mathematics > Statistics) ## Definition Finding the mean of a normal distribution from a probability statement involves determining the unknown mean $\mu$ when given that a normally distributed random variable $X \sim N(\mu, \sigma^2)$ satisfies a probability condition. This is typically done by standardizing the variable using the $z$-score: $ Z = \frac{X - \mu}{\sigma}, $ and then using the cumulative distribution function (CDF) of the standard normal distribution. ## Key Concepts - **Normal Distribution:** A continuous probability distribution defined by its mean $\mu$ and variance $\sigma^2$. - **Standardization:** Converting a normal variable to a standard normal variable $Z \sim N(0,1)$. - **Cumulative Distribution Function (CDF):** The function $\Phi(z)$ that gives the probability $P(Z \leq z)$. - **Inverse CDF:** Using $\Phi^{-1}(p)$ to find the $z$-value corresponding to a given probability $p$. ## Important Properties 1. **z-Score Transformation:** $Z = \frac{X - \mu}{\sigma}$ transforms $X \sim N(\mu, \sigma^2)$ to $Z \sim N(0,1)$. 2. **Probability Relationship:** For a given $x_0$, $P(X \leq x_0) = P\left(Z \leq \frac{x_0 - \mu}{\sigma}\right)$. 3. **Solving for $\mu$:** Rearranging the z-score formula allows one to solve for the unknown mean when $\sigma$, $x_0$, and the corresponding $z$-value are known. ## Essential Formulas - **z-Score Formula:** $ Z = \frac{X - \mu}{\sigma}. $ - **Rearranged for $\mu$:** $ \mu = X - \sigma Z. $ - **Probability Statement:** $ P(X \leq x_0) = \Phi\left(\frac{x_0 - \mu}{\sigma}\right). $ ## Core Examples 1. **Basic Example:** Let $X \sim N(\mu, 2^2)$ and suppose $P(X \leq 2.1) = 0.7088$. Standardize using: $ P\left(Z \leq \frac{2.1 - \mu}{2}\right) = 0.7088. $ From standard normal tables, $\Phi(0.55) \approx 0.7088$, so: $ \frac{2.1 - \mu}{2} = 0.55. $ Solving for $\mu$: $ 2.1 - \mu = 1.1 \quad \Longrightarrow \quad \mu = 1. $ 2. **Advanced Application:** If a different probability is given, the same method applies. For instance, if $P(X \leq x_0) = p$ with known $\sigma$, find $z = \Phi^{-1}(p)$ and solve: $ \frac{x_0 - \mu}{\sigma} = z \quad \Longrightarrow \quad \mu = x_0 - \sigma z. $ ## Related Theorems/Rules - **Standard Normal Distribution:** The properties and CDF $\Phi(z)$ of the standard normal distribution are used. - **Inverse Function Theorem:** Utilized when applying $\Phi^{-1}(p)$ to retrieve the corresponding $z$-score. ## Common Pitfalls - **Confusing Variance and Standard Deviation:** Remember that $\sigma$ is the standard deviation, not $\sigma^2$. - **Incorrect Standardization:** Failing to correctly subtract $\mu$ and divide by $\sigma$. - **Using the Wrong z-Value:** Ensure that the $z$-value corresponds to the correct probability from the standard normal table. - **Algebraic Errors:** Mistakes in rearranging the equation to solve for $\mu$. ## Related Topics - [[Normal Distribution]] - [[Standard Normal Distribution]] - [[Cumulative Distribution Function]] - [[Expected_Value_of_Discrete_Random_Variables]] - [[Discrete_Random_Variables]] - [[Estimating Mean and Variance of Grouped Data]] - [[Geometric Distribution]] - [[Addition Rule for Probability]] - [[Multiplication Law for Probability]] - [[Independent Events]] - [[Conditional Probability]] - [[Law of Total Probability]] - [[Law of Total Probability with Conditional Probability]] - [[Linear_Approximation]] - [[Limit of a Sequence]] - [[Residual]] - [[Linear Regression]] - **Standard Normal Distribution:** Understanding $Z \sim N(0,1)$ and its properties. - **z-Score Calculation:** The process of standardizing a normal variable. - **Parameter Estimation:** Techniques for estimating unknown parameters in probability distributions. ## Quick Review Questions 1. How do you transform a normally distributed variable $X \sim N(\mu, \sigma^2)$ into a standard normal variable $Z$? 2. Given $P(X \leq x_0) = p$ for a known $\sigma$, how do you solve for the unknown mean $\mu$? *** ![[Finding_the_Mean_of_a_Normal_Distribution_visualization]]