Week 7 - Continuous Random Variables and Probability Distributions

Probability Density Functions (PDFs): Explain and apply PDFs to determine probabilities for continuous random variables.
Continuous Probability Distributions: Describe and distinguish various continuous probability distributions and compute their key statistical measures.
Normal Distribution and Z-scores: Interpret and apply the normal distribution and Z-scores to solve real-world probability problems.

Random Variable: A numerical description of the outcome of an experiment.
Discrete Random Variable: A variable that may assume either a finite number of values or an infinite sequence of values.
Continuous Random Variable: A variable that may assume any numerical value in an interval or collection of intervals.

Guest Check-in Process:
- $X = \text{Time between guest arrivals (minutes)}$
- Interval: $\{X \geq 0\}$
Room Temperature Control:
- $X = \text{Temperature in a guest room (\degree C)}$
- Interval: $\{15 \leq X \leq 30\}$
Hotel Restaurant Operations:
- $X = \text{Amount spent by a guest on dinner (USD)}$
- Interval: $\{X \geq 0\}$
Energy Management:
- $X = \text{Daily electricity consumption (kWh)}$
- Interval: $\{X \geq 0\}$
Guest Satisfaction Tracking:
- $X = \text{Average satisfaction score (\%)}$
- Interval: $\{0 \leq X \leq 100\}$

Distribution	Density Function $f(x)$	Shape	Typical Use / Conditions	Expected Value $E[X]$	Variance $Var(X)$
Uniform	$f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$	Flat / rectangular	All outcomes equally likely within an interval.	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$
Normal	$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$	Bell-shaped, symmetric	Natural variation (room temp, spending, satisfaction).	$\mu$	$\sigma^2$
Exponential	$f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$	Right-skewed, decreasing	Time between arrivals or service calls.	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$
Z (Std. Normal)	$f(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}$	Normalize data to compare values $Z \sim N(0,1)$	Standardization (z-scores)	$0$	$1$
Student t	$f(x) = \frac{\Gamma(\frac{v+1}{2})}{\sqrt{v\pi} \Gamma(\frac{v}{2})} (1 + \frac{x^2}{v})^{-\frac{v+1}{2}}$	Normal with extra uncertainty (fatter tails)	Small samples, unknown variance; $X \sim t(v)$	$0$ (if $v > 1$ )	$\frac{v}{v-2}$ (if $v > 2$ )
Chi-Square	$f(x) = \frac{1}{2^{v/2}\Gamma(v/2)} x^{v/2 - 1} e^{-x/2}$	Measures variability (always positive)	Test of independence; $X \sim \chi^2(v)$	$v$	$2v$

Context: Weight of a cheeseburger.
Experiment: Making a cheeseburger.
Random Variable ( $X$ ): Weight of a cheeseburger.
Possible Values: $40 \leq x \leq 60$
Formula Representation:
- $f(x) = \frac{1}{20}$ for $40 \leq x \leq 60$
- $f(x) = 0$ otherwise.
Graphical Representation: A rectangular distribution with a height ( $h$ ) of $0.05$ (since $\frac{1}{60-40} = 0.05$ ), spanning the interval from $40$ to $60$ on the x-axis.

Notation: $X \sim N(\mu, \sigma^2)$
Parameters:
- $\mu$ : Mean (center of the distribution).
- $\sigma^2$ : Variance (spread of the distribution).

Scale: Puts data on a common scale (e.g., z-scores).
Comparison: Makes comparison across different units possible.
Interpretation: Enables interpretation relative to the mean and standard deviation.
Outlier Detection: Helps in identifying outliers.
Technical Requirement: Required for techniques like regression, Principal Component Analysis (PCA), and clustering.

Restaurant A:
- Average spend ( $\mu$ ) = $100\,CHF$
- Standard deviation ( $\sigma$ ) = $20\,CHF$
- Distribution: $X \sim N(100, 20^2)$
Restaurant B:
- Average spend ( $\mu$ ) = 20\,\text{\euro}
- Standard deviation ( $\sigma$ ) = 5\,\text{\euro}
- Distribution: $X \sim N(20, 5^2)$
Standardization Formula:
- $Z = \frac{X - \mu}{\sigma}$
- The resulting $Z$ follows a standard normal distribution: $Z \sim N(0, 1)$ .
- Units of measurement correspond to a standard deviation of $1$ .

Empirical Rule (68-95-99.7 Rule):
- Approximately 68% of data falls within $1$ standard deviation of the mean ( $\pm 1\sigma$ ).
- Approximately 95% of data falls within $2$ standard deviations of the mean ( $\pm 2\sigma$ ).
- Approximately 99.7% of data falls within $3$ standard deviations of the mean ( $\pm 3\sigma$ ).
Area Breakdown (Symmetric):
- Center to $1\sigma$ : $34\%$
- $1\sigma$ to $2\sigma$ : $13.5\%$
- Beyond $2\sigma$ : $2.35\%$

Cumulative Probability $P(X \leq x)$ :
- =NORM.DIST(x; \mu; \sigma; 1)
Right-Tail Probability $P(X > x)$ :
- =1 - NORM.DIST(x; \mu; \sigma; 1)
Interval Probability $P(x_0 < X \leq x_1)$ :
- =NORM.DIST(x1; \mu; \sigma; 1) - NORM.DIST(x0; \mu; \sigma; 1)

Distribution Check: $Z \sim N(0, 1)$
Standard Normal Probability:
- =NORM.S.DIST(z; 1) which is equivalent to =NORM.DIST(z; 0; 1; 1)
Standard Normal Inverse:
- =NORM.S.INV(cumulative_probability) which is equivalent to =NORM.INV(cumulative_probability; 0; 1)