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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 – We’re familiar with the normal distribution already, but we’ll learn about distributions with other “shapes” later 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) Discrete RV’s & Distributions • Probability function of a discrete RV – Called probability mass function (pmf) – See graph (Actex page 112) 0 p(x) 1 p(x) 1 x P(X A) p(x) P(A) x A 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) Continuous RV’s & Distributions • Properties f (x) 0 f (x)dx 1 P[X (a,b)] P(a X b) b f (x)dx a b F(b) P( x b) b a f (x)dx F(b) F(a) f (x)dx 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” 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! Conditional Distributions • Recall: • Suppose : – fX(x) is “marginal density” of X f X (x) – A is an event f X |A (x | A) P(A) – 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