5. Continuous Probability Distributions

1. Uniform Distribution

• Use: Every value in a range is equally likely.
• Example (Random Number Generator):
• A computer generates a random number $X$ between 0 and 10.
• Question: Probability the number is less than 3?
• Range width = 10. Target width = 3.
• Calculation:

$$ P(0 < X < 3) = \frac{\text{Target Width}}{\text{Total Width}} = \frac{3}{10} = 0.3 $$

2. Exponential Distribution

• Use: Time until an event happens (e.g., failure of a machine, arrival of a bus).
• Example (Battery Life):
• The average life of a battery is 10 hours. Rate $\lambda = 1/10 = 0.1$.
• Question: Probability the battery lasts less than 5 hours?
• Calculation: CDF Formula for Exponential is $F(x) = 1 - e^{-\lambda x}$.

$$ P(X < 5) = 1 - e^{-0.1(5)} $$ $$ P(X < 5) = 1 - e^{-0.5} \approx 1 - 0.606 = 0.394 $$ (39.4% chance the battery dies before 5 hours).

3. Normal Distribution

• Use: Bell curve. Natural phenomena (height, IQ, errors).
• Example (Student Heights):
• Average height ($\mu$) = 170 cm. Standard Deviation ($\sigma$) = 10 cm.
• Question: What is the probability a student is shorter than 160 cm?
• Step 1: Find Z-score.

$$ Z = \frac{X - \mu}{\sigma} = \frac{160 - 170}{10} = \frac{-10}{10} = -1 $$

• Step 2: Look up Z-table or use rule.
• Z = -1 means the value is 1 standard deviation below the mean.
• According to the Empirical Rule (68-95-99.7 rule), roughly 68% of data is within $\pm 1\sigma$.
• This leaves 32% in the tails.
• Since the curve is symmetric, the lower tail (below -1) is roughly $32\% / 2 = 16\%$.
• Result: Probability is approximately 0.16 (or 16%).