Lecture Notes on Normal and Exponential Distributions

Recap of Normal Distributions

Probability Density Function (PDF):
- Formula: ( f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} )
- Parameters: ( \mu ) (mean) and ( \sigma^2 ) (variance)
- Symmetric distribution: mean = median = mode
Standard Normal Variable (Z):
- Formula: ( Z = \frac{X - \mu}{\sigma} )
- Mean = 0, Variance = 1
- Cumulative Distribution Function (CDF): ( \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{y^2}{2}} dy )
- Symmetry: ( \Phi(-x) = 1 - \Phi(x) )

Sum of Normal Random Variables:
- If ( X_i ) are independent normal variables with means ( \mu_i ) and variances ( \sigma_i^2 ), then ( \sum X_i ) is also normal.
- Mean of ( \sum X_i = \sum \mu_i )
- Variance of ( \sum X_i = \sum \sigma_i^2 )

Rainfall Probability (2 Years):
- Yearly rainfall: Normal with ( \mu=12 \text{cm}, \sigma=3 \text{cm} )
- Probability of total rainfall over 2 years > 25cm: Use sum and convert to standard normal.
Yearly Rainfall Comparison:
- Probability of current year exceeding next year by > 3cm.

PDF and CDF:
- ( f(x) = \lambda e^{-\lambda x} ) for ( x \geq 0 )
- CDF: ( F(x) = 1 - e^{-\lambda x} )
- Memoryless Property: ( P(X > t+s | X > t) = P(X > s) )

( M(t) = \int_0^\infty e^{tx} \lambda e^{-\lambda x} dx = \frac{\lambda}{\lambda - t} )
Mean: ( E(X) = \frac{1}{\lambda} )
Variance: ( Var(X) = \frac{1}{\lambda^2} )

Battery Life:
- Exponentially distributed with average life of 10,000 km.
- Probability of lasting > 5,000 km: Use memoryless property.