Statistics Chapter 5: Continuous and Normal Random Variables

Continuous random variable

can assume any numerical value within some interval or intervals

Ex: Randomly choose a battery from a production line and record its lifetime

Probability Distributions for Continuous Random Variables

• the graph of the probability distribution is a smooth curve called: probability density function (f(x))

• there are an infinite number of outcomes

Mean of a numerical variable x (μ)

describes where the probability distribution of x is centered

Interpretation: if you randomly pick…, then in the long run, on average, the….will be μ = …

Standard deviation of a numerical variable x (σ)

describes variability in the probability distribution

1. When σ is close to 0, observed values of x will tend to be close to the mean value

2. When σ is large, there will be more variability in observed values

The Uniform Distribution

x can take on any value between c and d with equal probability

Normal Random Variable

has a bell shaped probability distribution and is one of the most commonly observed continuous random variables

Normal random variable properties

• plays an important role in the science of statistical inference, turns to be a very reasonable approximation in real life problems

• most important: Central Limit Theorem

Normal Distribution

a continuous random variable X is said to have a normal distribution with mean (μ) and variance (σ²) greater than 0 if its probability distribution looks like f(x)

Two characteristics values (numbers) completely determine a normal distribution

1) Mean - μ

2) Standard deviation - σ

Standard (or standardized) Normal Distribution

a normal distribution with μ = 0 and σ² = 1

Steps to find the probability for a Normal Random Variable

1) Formulate the problem

2) Sketch the normal distribution and indicate the mean of the random variable X and then shade the area corresponding to the probability you want to find

3) Convert the X variables into Z values

4) Put both sets of values on the sketch, z below x

5) Use Standard Normal Table to find the desired probabilities