Exam Questions - Probability density functions and cumulative distribution functions

Exam Questions - Probability density functions and cumulative distribution functions

Understanding Probability Density Functions (PDFs)

  • Probability Density Functions, represented as f(x), describe the likelihood for a continuous random variable to take on a certain value.
  • In relation to PDFs, it’s essential that the function is equal to or greater than zero for all x values, namely f(x) ≥ 0.
  • The area under the curve of a PDF represents the probability; therefore, the integral of a PDF over its entire domain is equal to 1.
  • In calculus terms, this leads us to a key property of PDFs: ∫ f(x) dx = 1 (from ∞ to -∞).

Calculation of PDFs and Expected Values

  • To calculate the probability that a random variable falls within a certain range, calculate the integral of the PDF over that range.
  • The expected value (mean) of a continuous random variable is calculated by integrating x times the PDF over its entire domain: E(X) = ∫x f(x) dx.
  • The variance, which measures the spread of the distribution, is calculated as: Var(X) = E[(X - µ)^2] = E(X^2) - (E(X))^2.

Understanding Cumulative Distribution Functions (CDFs)

  • Cumulative Distribution Functions, or F(x), provide the cumulative probability up to a certain value.
  • A CDF effectively tells you the probability that a random variable is less than or equal to a certain value.
  • Unlike a PDF, a CDF is monotonically increasing; that is, it doesn’t decrease or stay constant - it only increases or stays the same.

Calculation of CDFs

  • The CDF at a certain value x is calculated by finding the integral of the PDF up to x, that is: F(x) = ∫ f(t) dt (from -∞ to x).
  • Alternatively, if you have a discrete distribution, you can derive the CDF by adding up the individual probabilities up to the chosen value.

Connection between PDFs and CDFs

  • PDFs and CDFs are closely related - the CDF is the integral of the PDF, and the PDF is the derivative of the CDF.
  • Graphically, the y-value of the PDF at a certain x-value represents the slope of the CDF at that same x-value.
  • Remember that this applies only when the random variable is continuous. For discrete random variables, PDFs and CDFs are related in different ways.

Application of PDFs and CDFs

  • Both PDFs and CDFs are crucial tools in understanding and describing probability distributions in statistics.
  • They can be used to calculate the probabilities of different outcomes or to determine statistical metrics like the mean and variance.
  • Understanding the application of PDFs and CDFs can significantly aid in solving problems related to real-world situations, such as calculating risks or optimising operations.

Potential Pitfalls When Working with PDFs and CDFs

  • It’s important not to confuse PDFs and CDFs, remembering that while the former shows the probability at a particular value, the latter provides cumulative probability.
  • The properties of certain distributions (normal, binomial, exponential, etc.) often have their own unique PDFs and CDFs - be familiar with these for commonly used distributions.
  • Beware of the bounds of integration when computing probabilities or expected values - it’s essential to respect the domain of the random variable in question.
  • Always verify that any function you’re using as a PDF satisfies the properties of PDFs: namely, it must be non-negative, and its integral must be 1.