Introduction

Chapter 4

Continuous random variables and their probability distributions

Introduction

Up to this point, we have limited our discussion to probability distributions of discrete random variables. Recall that a discrete random variable takes on only some isolated values, usually integers representing a count. We now turn our attention to the probability distribution of a continuous random variable- one that can ideally assume any value in an interval. Variables measured on an underlying continuous scale, such as weight, strength, life length, and temperature, have this feature.

Figure 4.1 displays the histogram and

polygon for some continuous data set.

The smoothed polygon is an approximation

of the probability distribution curve of the continuous random variable X. The probability distribution curve of a continuous random variable is also called

its probability density function.

The probability density function, denoted by possesses the following characteristics:

1. for all x.

2. The area under the probability density function over all possible values of the random variable X

is equal to 1.

3. Let a and b be two possible values of the random variable X, with . Then the probability that X lies between a and b is the area under the density function between a and b. (Fig.4.2)

4. The cumulative distribution function is the area under the probability density function up to

where is the minimum value of the random variable X.

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