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Section 4 – Random Variables and Probability Distributions

Section 4 – Random Variables and Probability Distributions. Random Variables. Capital X denotes a random variable Based on the distribution , there are different probabilities for each value that X could take on

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Section 4 – Random Variables and Probability Distributions

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  1. Section 4 – Random Variables and Probability Distributions

  2. Random Variables • Capital X denotes a random variable • Based on the distribution, there are different probabilities for each value that X could take on • We’re familiar with the normal distribution already, but we’ll learn about distributions with other “shapes” later

  3. Discrete vs. Continous • Discrete RV: X can only be from a finite set of values (usually integers or only some integers) • Ex: The number of customers that walks in • Probability Mass Function (pmf) • Continuous RV: X can be a value anywhere in a range of numbers • Ex: The exact time a customer walks in • Probability Density Function (pdf)

  4. Discrete RV’s & Distributions • Probability function of a discrete RV • Called probability mass function (pmf) • See graph (Actex page 112)

  5. Continuous RV’s & Distributions • Probability function of a continuous RV • Called a probability density function (pdf) • See graph (Actex p113) f(x) = probability density function (pdf) F(x) = P(X<=x) = cumulative probability density function (cdf) S(x) = 1- F(x) = P(X>x)

  6. Continuous RV’s & Distributions • Properties

  7. Greater Than vs. Greater Than or Equal To • Discrete • The distinction is important (there is likely some probability on the integer included or left out) • Continuous • The distinction is NOT important • For any value x, f(x) = 0 • No one point has any probability (integrating from a to a is 0) so there will be no probability left out by not including the “edge”

  8. Mixed Distributions • Mixed Distribution: A random variable’s distribution where it has discrete probabilities for some interval and continuous on another • Basically continous and discrete • f(x) is a piecewise function • Total probability still sums to 1 • Note: these are not “mixtures of distributions” – that topic will come much later, but remember to study both!

  9. Conditional Distributions • Recall: • Suppose : • fX(x) is “marginal density” of X • A is an event • We will go into more detail about calculating the marginal density in our next review session. Note: if X is not an outcome in event A, fX|A(x|A)=0

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