Notes 6

Notes 6
Statistics
Continuous Probability Distributions
Probability Density Function [f(x)] –area under the graph of f(x) gives probability
Uniform Probability Distribution
!
for a ≤ x ≤ b f(x) = !!!
0 elsewhere
!!!
(!!!)!
E(x) = ! , Var(x) = !
Normal Probability Distribution
Most important continuous probability distribution
Many applications: heights, weights, rainfall, test scores
Used extensively in statistical inference
Shape of normal distribution is bell-shaped
Characteristics:
1. All normal distributions differentiated by µμ and σ
2. Highest point is µμ (also median and mode)
3. Mean can be any value (negative, zero or positive)
4. Symmetric around µμ. Tails extend to infinity.
5. σ determines how flat and wide curve is (larger value flatter curve)
6. Probabilities given by area under curve. Total area is 1. Half area is .5.
7. Common percentages: 68.3% within 1, 95.4% within 2, 99.7% within 3
Standard Normal Probability Distribution
Normal distribution with µμ = 0 and σ = 1
Random variable is written as z
Three types of probabilities to compute:
1. Probability that z will be less than or equal to a value
2. Probability that z will be between two values
3. Probability that z will be greater than or equal to a value
!!!
Converting to Standard Normal: z = !
Exponential Probability Distribution
Applications: time between customers at a car wash, time to load a truck
!
f x = ! e!!/!
P(x≤x0) = 1 - e!!!

!

Examples
1. x = flight time from Chi to NY, x is uniformly distributed between 120 and 140
| P(120≤ x ≤130), P(x>135), P(x=130)
2. z is standard normal random variable | P(z≤1), P(-.5≤z≤1.25), P(z≥1.58)

3. Normal distribution with µμ = 10, σ = 2 | P(10≤x≤14)
4. x = time to load a truck, µμ =15 | P(6≤x≤18)
Practice
1. x unif dist bet 1 and 1.5: graph, P(x=1.25), P(1≤x≤1.25), P(1.20<x<1.5)
2. z is standard normal random variable | P(z≤1.2), P(-1.57≤z≤0), P(z≤-.23)
3. Normal distribution with µμ = 20, σ = 5 | P(17≤x≤23)
4.