Econ 120A - Notes on Continuous Random Variables

Continuous Random Variables

Continuous Probability Distributions

• Key topics:
  • Continuous Random Variables
  • Uniform Distribution
  • Normal Distribution
  • Standard Normal Distribution

Continuous Random Variables

• 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}.
• P(x1 \leq X \leq x2) = \text{Base} \times \text{Height} = \frac{x2 - x1}{b - a}