4.1 Definition and properties of continuous random variables

Continuous random variables are the heart of probability theory. They can take any value within a range, unlike discrete variables that only have specific values. This flexibility allows us to model real-world phenomena more accurately.

Understanding continuous random variables is crucial for grasping probability distributions. We'll learn about probability density functions, cumulative distribution functions, and how to calculate probabilities and expected values for continuous variables.

Continuous Random Variables

Definition and Characteristics

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• A continuous random variable can take on any value within a specified interval or range
• The probability of a continuous random variable taking on any single specific value equals zero since there are infinitely many possible values within the range
• Continuous random variables are represented by a probability density function (PDF) which describes the relative likelihood of the variable taking on a particular value (normal distribution, exponential distribution)
• The cumulative distribution function (CDF) of a continuous random variable gives the probability that the variable takes a value less than or equal to a given value
  • The CDF is the integral of the PDF from negative infinity to the given value
• The expected value (mean) and variance of a continuous random variable can be calculated using the PDF
  • The expected value is the weighted average of all possible values, where the weights are the probabilities of each value occurring
  • The variance measures the average squared deviation from the mean, indicating the spread of the distribution

Properties and Implications

• The probability density function (PDF) of a continuous random variable is non-negative everywhere, and the total area under the PDF curve equals 1
  • This property ensures that the probabilities of all possible outcomes sum to 1
• The probability of a continuous random variable falling within a specific range equals the area under the PDF curve over that range
  • To find the probability, integrate the PDF over the desired range
• The cumulative distribution function (CDF) of a continuous random variable is a non-decreasing function with values between 0 and 1
  • As the value of the random variable increases, the CDF either increases or remains constant, but never decreases
• The CDF can be used to calculate probabilities for a continuous random variable by finding the difference between the CDF values at the upper and lower limits of the desired range
  • P(a ≤ X ≤ b) = CDF(b) - CDF(a), where X is the continuous random variable and a and b are the lower and upper limits, respectively
• The variance of a continuous random variable measures the average squared deviation from the mean and is calculated using the PDF and the expected value
  • A higher variance indicates a greater spread in the distribution, while a lower variance suggests the values are more concentrated around the mean (standard deviation is the square root of variance)

Discrete vs Continuous Random Variables

Key Differences

• Discrete random variables can only take on a countable number of distinct values, while continuous random variables can take on any value within a specified range
  • Examples of discrete random variables: number of defective items in a batch, number of customers in a queue
  • Examples of continuous random variables: height of students in a class, time until a light bulb fails
• The probability mass function (PMF) describes the probability distribution of a discrete random variable, while the probability density function (PDF) is used for continuous random variables
  • The PMF gives the probability of a discrete random variable taking on a specific value
  • The PDF represents the relative likelihood of a continuous random variable falling within a particular range
• For discrete random variables, the probability of a specific value can be non-zero, while for continuous random variables, the probability of any single value is always zero
  • In a discrete distribution (rolling a die), each possible outcome has a non-zero probability
  • In a continuous distribution (selecting a random real number between 0 and 1), the probability of selecting any single value is zero

Cumulative Distribution Functions

• The cumulative distribution function (CDF) for a discrete random variable is a step function, while the CDF for a continuous random variable is a continuous function
  • For a discrete random variable, the CDF increases by the probability of each value at the corresponding point
  • For a continuous random variable, the CDF is a smooth, continuous curve that represents the area under the PDF up to a given point
• The CDF of a discrete random variable can be used to calculate probabilities by summing the probabilities of all values less than or equal to a given value
  • P(X ≤ x) = sum of P(X = k) for all k ≤ x, where X is the discrete random variable and x is the given value
• The CDF of a continuous random variable can be used to calculate probabilities by finding the difference between the CDF values at the upper and lower limits of the desired range
  • P(a ≤ X ≤ b) = CDF(b) - CDF(a), where X is the continuous random variable and a and b are the lower and upper limits, respectively

Properties of Continuous Random Variables

Probability Density Function (PDF)

• The PDF of a continuous random variable is a function that describes the relative likelihood of the variable taking on a particular value
  • The PDF is denoted by f(x), where x is the value of the random variable
• The PDF is non-negative everywhere, meaning f(x) ≥ 0 for all values of x
  • This property ensures that probabilities are always non-negative
• The total area under the PDF curve equals 1, representing the sum of all probabilities
  • This property ensures that the probabilities of all possible outcomes sum to 1
  • Mathematically, the integral of f(x) from negative infinity to positive infinity equals 1: $\int_{-\infty}^{\infty} f(x) dx = 1$
• The probability of a continuous random variable falling within a specific range equals the area under the PDF curve over that range
  • To find the probability of X falling between a and b, integrate the PDF from a to b: $P(a \leq X \leq b) = \int_{a}^{b} f(x) dx$

Expected Value and Variance

• The expected value (mean) of a continuous random variable is the weighted average of all possible values, where the weights are the probabilities of each value occurring
  • The expected value is denoted by E(X) or μ and is calculated using the PDF: $E(X) = \int_{-\infty}^{\infty} x \cdot f(x) dx$
  • The expected value represents the average value of the random variable over a large number of trials
• The variance of a continuous random variable measures the average squared deviation from the mean and is calculated using the PDF and the expected value
  • The variance is denoted by Var(X) or σ^2 and is calculated using the formula: $Var(X) = \int_{-\infty}^{\infty} (x - \mu)^2 \cdot f(x) dx$
  • A higher variance indicates a greater spread in the distribution, while a lower variance suggests the values are more concentrated around the mean
• The standard deviation is the square root of the variance and is denoted by σ
  • The standard deviation has the same units as the random variable and provides a measure of the typical distance between a value and the mean