Probability Distributions

Probability Density Function

Transformation of PDF

Call  the probability distribution function of the random variable . For a transformation of the form , we have the following holds:In other words,

e.g. Suppose . For , we have  and then . Moreover , then the result is .

Cumulative Distribution Function

Def Cumulative Distribution Function The cumulative distribution function (cdf) of a real random variable , evaluated at , is the probability that will take a value less than or equal to :

Prop

• is right-continuous monotone increasing
• and Proof

Prop Clearly by definition of cumulative distribution function, we have

Prop The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating:

Expected Value

Expected Value

Let be a probability space, and be a measurable real-valued random variable. The expected value of , denoted , is defined as the Lebesgue integral of with respect to the probability measure :