Section 4.3 Random Variables

probability model:

• describes the possible outcomes of a chance process and the likelihood that those outcomes will occur

random variable:

• takes numerical values that describe the outcomes of some chance process

probability distribution:

• of a random variable gives its possible values and their probabilities

Discrete Random Variables:

• There are two main types of random variables: discrete and continuous

• If you can find a way to list all possible outcomes for a random variable and assign probabilities to each one, then you have a discrete random variable

• A discrete random variable X takes a fixed set of possible values with gaps
  between. The probability distribution of a discrete random variable X lists the
  values xi and their probabilities pi:

• Value: x1 ×2 ×3

• Probability: p1 p2 p3

• The probabilities pi must satisfy two requirements:

  • 1. Every probability pi is a number between 0 and 1.

  • 2. The sum of the probabilities is 1

  • To find the probability of any event, add the probabilities pi of the particular values xi that make up the event

Continuous Random Variables:

• Discrete random variables commonly arise from situations that involve counting something. Situations that involve measuring something often result in a continuous random variable.

• A continuous random variable Y takes on all values in an interval of numbers. The probability distribution of Y is described by a density curve.

• The probability of any event is the area under the density curve and above the values of Y that make up the event.

• Unlike the probability model of a discrete random variable X that assigns a probability between 0 and 1 to each possible value of X, a continuous random variable Y has infinitely many possible values.

• All continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability
  

Continuous Probability Models:

• A continuous probability model assigns probabilities as areas under a density curve.

• The area under the curve and above any range of values is the probability of an outcome in that range
  

Normal Probability Models:

• Normal distributions are probability models

• Probabilities can be assigned to intervals of outcomes using the standard Normal probabilities in Table A

• We standardize Normal data by calculating z-scores so that any Normal curve N(μ, s) can be transformed into the standard Normal curve N(0, 1)

• z = (x-h (upsidedown) / o hat

• Often, the density curve used to assign probabilities to intervals of outcomes is the Normal curve