Examples 5 tosses of a coin. This is a random variable: The number of heads ■This is not: 2005 005 2005
Examples ◼ 5 tosses of a coin. ◼ This is a random variable: The number of heads ◼ This is not:
Main Concepts Related to Random Variables We can associate with each random variable certain“averages”of interest,such as the mean and the variance. A random variable can be conditioned on an event or on another random variable. Notion of independence of a random variable from an event or from another random variable. We'll talk about all these in this lecture
Main Concepts Related to Random Variables ◼ We can associate with each random variable certain “averages” of interest, such as the mean and the variance. ◼ A random variable can be conditioned on an event or on another random variable. ◼ Notion of independence of a random variable from an event or from another random variable. ◼ We’ll talk about all these in this lecture
Discrete Random Variable A random variable is called discrete if its range is either finite or countably infinite. Example.Two rolls of a die. The sum of the two rolls. The number of sixes in the two rolls. The second roll raised to the fifth power
Discrete Random Variable ◼ A random variable is called discrete if its range is either finite or countably infinite. ◼ Example. Two rolls of a die. ❑ The sum of the two rolls. ❑ The number of sixes in the two rolls. ❑ The second roll raised to the fifth power
Continuous random variable Example.Pick a real number a and associate to it the numerical value a2. ■ The random variable a2 is continuous,not discrete. We'll talk about continuous random variables later. The following random variable is discrete: 了1 a> 0 ( a=0. a<0
Continuous random variable ◼ Example. Pick a real number 𝑎 and associate to it the numerical value 𝑎 2 . ◼ The random variable 𝑎 2 is continuous, not discrete. ◼ We’ll talk about continuous random variables later. ◼ The following random variable is discrete: 𝑠𝑖𝑔𝑛 𝑎 = ቐ 1 𝑎 > 0 0 𝑎 = 0 −1 𝑎 < 0
Discrete Random Variables:Concepts A discrete random variable is a real-valued function of the outcome of a discrete experiment. A discrete random variable has an associated probability mass function (PMF),which gives the probability of each numerical value that the random variable can take. A function of a discrete random variable defines another discrete random variable,whose PMF can be obtained from the PMF of the original random variable
Discrete Random Variables: Concepts ◼ A discrete random variable is a real-valued function of the outcome of a discrete experiment. ◼ A discrete random variable has an associated probability mass function (PMF), which gives the probability of each numerical value that the random variable can take. ◼ A function of a discrete random variable defines another discrete random variable, whose PMF can be obtained from the PMF of the original random variable