← Stat 110: Probability

Unit 5: Continuous Random Variables

Probability spread over a continuum instead of piled on points — PDFs, the Uniform's universality, the Normal curve, and the Exponential's total lack of memory.

9 topics · 32 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

5.1 PDFs: Probability Density Instead of Probability Mass

  1. How it worksHow does integrating a PDF over an interval give the probability of landing in that interval?
  2. Worked exampleFind P(0.2 less than X less than 0.5) for a random variable uniform on the interval from 0 to 1.
  3. Common mistakeWhy isn't a PDF's height at a point the probability of the random variable equaling that point?
  4. Going furtherWhat happens to a PDF's total area once you integrate it across its entire range?
Your turn1 of 2
Find P(2 < X < 5) for a random variable uniform on the interval from 0 to 19.

5.1 The CDF-PDF Relationship

  1. How it worksHow does the fundamental theorem of calculus link a CDF's slope to its PDF's height?
  2. Worked exampleFind the PDF of a random variable whose CDF is F(x) equal to x2 for x between 0 and 1.
  3. Going furtherWhat happens to the CDF-PDF relationship at a point where the PDF has a sudden jump in height?
Your turn1 of 2
A random variable has CDF F(x) = x2 for x between 0 and 1. Write its PDF as a function of x.

5.2 The Uniform Distribution

  1. How it worksHow does a constant PDF height over an interval turn into the uniform distribution's formula?
  2. Worked exampleFind the probability a bus arrives in the first 4 minutes of a 10-minute uniformly random window.
  3. Common mistakeWhy isn't a uniform distribution on an interval the same as saying every real number is equally likely?
Your turn1 of 2
Find the probability a bus arrives in the first 6 minutes of a 45-minute uniformly random window.

5.2 The Universality of the Uniform

  1. How it worksHow does pushing a uniform random variable through an inverse CDF land it distributed as the target?
  2. Worked exampleFind the transformation that turns a uniform random number on 0 to 1 into an exponential random variable with rate 2.
  3. Common mistakeWhy doesn't universality of the uniform require the target distribution to be continuous everywhere?
  4. Ties togetherHow does the universality of the uniform connect to how a computer actually simulates a Normal draw?
Your turn1 of 2
Write the transformation that turns a uniform random number u on 0 to 1 into an exponential random variable with rate lam, as a function of u.

5.3 The Normal Distribution

  1. How it worksHow do the mean and standard deviation alone determine a Normal curve's entire shape?
  2. Common mistakeWhy doesn't a Normal distribution assign exactly zero probability to values far out in its tails?
  3. Worked exampleFind P(X less than 70) for a Normal random variable with mean 65 and standard deviation 5.
  4. Going furtherWhat happens to a Normal curve's shape as its standard deviation shrinks toward zero?
Your turn1 of 2
A Normal random variable has mean 78 and standard deviation 11. Find its z-score at the value 113.

5.3 Standardization and Z-Scores

  1. How it worksHow does subtracting the mean and dividing by the standard deviation turn any Normal into the standard Normal?
  2. Worked exampleFind the z-score of a value of 82 from a Normal distribution with mean 70 and standard deviation 8.
  3. Going furtherWhat happens to a z-score when the original distribution's standard deviation doubles?
Your turn1 of 2
Find the z-score of a value of 99 from a Normal distribution with mean 72 and standard deviation 19.

5.4 The Law of the Unconscious Statistician (LOTUS)

  1. How it worksHow does LOTUS let you plug g(x) straight into the original PDF and integrate?
  2. Worked exampleFind E[X2] for a random variable uniform on the interval from 0 to 2 using LOTUS.
  3. Common mistakeWhy isn't E[g(X)] generally equal to g(E[X])?
Your turn1 of 2
Find E[X2] for a random variable uniform on the interval from 0 to 15, using LOTUS.

5.5 The Exponential Distribution

  1. How it worksHow does the exponential distribution arise as the continuous limit of a geometric waiting time?
  2. Common mistakeWhy doesn't an exponential random variable's rate parameter equal its mean instead of 1 over the mean?
  3. Worked exampleFind the probability a lightbulb with an average lifetime of 1000 hours survives past 1500 hours.
  4. Going furtherWhat happens to the exponential distribution's PDF as the rate parameter lambda increases?
Your turn1 of 2
Find the probability a lightbulb with an average lifetime of 2241 hours survives past 3280 hours.

5.5 The Memoryless Property

  1. How it worksHow does the exponential distribution's memorylessness fall directly out of its formula?
  2. Worked exampleFind P(a lightbulb with rate 1/1000 survives another 200 hours, given it has already survived 800 hours).
  3. Common mistakeWhy isn't the memoryless property true of a Normal or Uniform random variable's remaining time?
  4. Ties togetherHow does the exponential distribution's memoryless property connect to the geometric distribution's version?
Your turn1 of 2
A lightbulb's lifetime is exponential with rate 0.00154 per hour. By the memoryless property, find P(it survives another 342 hours given it has already survived 939 hours).