Definition: Continuous Random Variables can take on any value within a given interval on the real number line or within a collection of intervals.
The number of possible values for a continuous random variable is uncountable.
Examples of Continuous Random Variables (Table 5.2):
Customer visits a web page:
Random Variable (X): Time customer spends on the web page in minutes.
Possible Values: X \geq 0
Fill a soft drink can (max capacity = 12.1 ounces):
Random Variable (X): Number of ounces.
Possible Values: 0 \leq x \leq 12.1
Test a new chemical process (min temperature = 150°F; max temperature = 212°F):
Random Variable (X): Temperature when the desired reaction takes place.
Possible Values: 150 \leq x \leq 212
Invest $10,000 in the stock market:
Random Variable (X): Value of investment after one year.
Possible Values: X \geq 0
Probability Distribution of a Continuous Random Variable
The probability of a continuous random variable assuming a particular value is considered zero.
Instead of discussing the probability at a specific value, we analyze the probability of the variable falling within a given interval.
Histogram and Probability Density Function
Histograms can represent frequencies of data within intervals.
The probability density function (PDF) is used to find the probability of a continuous random variable falling within an interval.
Calculating Probability with PDF
To find the probability of a continuous random variable falling within an interval, we calculate the area under the probability density function (pdf) over that interval.
Since the probability of a continuous random variable assuming a single specific value is zero, P(X=c) = P(X=d) = 0.
Therefore, P(c \leq X \leq d) = P(c < X < d)
The probability is calculated by integrating the probability density function f(x) from a to b:
P(a < X < b) = \int_{a}^{b} f(x) dx
Summary: Probability Density Function (pdf)
Every continuous random variable X has a probability density function f(x).
The total area under the curve of the pdf is always 1.
The probability of a continuous random variable X assuming a value between c and d, P(c \leq X \leq d), is the area under the curve (the pdf) between the values c and d.
Since P(X=c) = P(X=d) = 0, then P(c \leq X \leq d) = P(c < X < d)
Uniform Probability Distribution
A random variable is uniformly distributed when the probability is proportional to the interval’s length.
a = smallest value the variable can assume.
b = largest value the variable can assume.
The uniform probability density function is defined as:
f(x) = \begin{cases} \frac{1}{(b-a)} & \text{for } a \leq x \leq b \ 0 & \text{elsewhere} \end{cases}
Uniform Distribution Probability Calculation
To calculate the probability that X falls between x1 and x2, determine the area of the rectangle with base x2 - x1 and height \frac{1}{b - a}.