Stat 226 - Lecture 7

02/04/14

Probability...It's as easy as flipping a coin!

Probability made easy

  • What's the probability that a coin lands heads-up?
  • Assuming it's a fair coin, the probability is 0.5.
  • What's the probability that a coin lands tails-up?
  • We just fully described a probability distribution function (pdf)!

  • In essence, a pdf assigns probability to value(s) of a random variable.

  • Technically, value(s) of a random variable define event(s) and each event has a probability.

  • Suppose I flip a coin twice. What are the different possible events?

  • Let \( X \) be the # of heads after two flips. What event(s) correspond to \( X=1 \)?

  • 2 out of 4 (equally likely) events lead to \( X=1 \), so \( P(X=1) = 0.5 \).

    • What are all the different possible values \( X \)?
    • What is the probability \( X \) takes on each value?

  • probability distribution function for \( X \):

    P(X=0) P(X=1) P(X=2)
    0.25 .5 0.25

    • Since \( X \) takes on a small # of values, it is a discrete random variable (think categorical).
    • From now on, we study continuous random variables (can you guess the difference?)
    • Continuous random variables can take on an infinite number of values (think quantitative)!
    • Events are defined by intervals: \( X \leq 0 \), not \( X=0 \)
    • IMPORTANT: probabilities are always between 0 and 1! pdfs always add to 1!

    • Back to normal distributions

    • The Normal distribution is a special type of pdf.
    • A normal random variable is a continous random variable whose pdf is symmetric and bell-shaped.
    • If \( X \) is a normally distributed random variable, it has a mean \( \mu \) and variance \( \sigma^2 \).
    • More succinctly, \( X \sim N(\mu, \sigma^2) \)

    • The Empirical Rule

    • Given \( X \sim N(\mu, \sigma^2) \), the empirical rule says:
    • The probability that \( X \) is within one standard devation of the mean is 0.68:
    • That is, \( P(\mu - \sigma < X < \mu + \sigma ) = 0.68 \).
    • The probability that \( X \) is within two standard devations of the mean is 0.95.
    • That is, \( P(\mu - 2\sigma < X < \mu + 2\sigma ) = 0.95 \).
    • The probability that \( X \) is within three standard devations of the mean is 0.997.
    • That is, \( P(\mu - 3\sigma < X < \mu + 3\sigma ) = 0.997 \).

    Your Turn

    (1) Let \( X \sim N(100, 50^2) \). Find the following probabilities:

    • \( P(X < 100) \)
    • \( P(X < 150) \)
    • \( P(100 < X < 150) \)
    • \( P(X < 0) \)

    (2) Suppose \( Z \sim N(0, 1) \). Find the following probabilities:

    • \( P(Z < 0) \)
    • \( P(Z < 1) \)
    • \( P(0 < Z < 1) \)
    • \( P(Z < -2) \)

    Standard Normal Distribution

    • The standard normal distribution is a normal distribution with mean 0 and variance 1.
    • We typically use \( Z \) to represent a standard normal distribution.
    • It turns out that we can transform any \( X \sim N(\mu, \sigma) \) into \( Z \sim N(0,1) \) using a z-score:
    • \[ z = \frac{x-\mu}{\sigma} \]
    • Example(s) on board.