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What does it mean to condition on a random variable?
Conditioning on an event (such as a particular specification of a random variable) means that this event is treated as being known to have occurred. This still allows us to specify conditioning on an event {Y=y} where the actual value y is an algebraic variable that falls within some range.
Can a random variable take on any value?
A random variable can be either discrete or continuous. Discrete random variables take on a countable number of distinct values. Continuous random variables can represent any value within a specified range or interval and can take on an infinite number of possible values.
What does it mean to condition on a variable?
Conditioning on a variable involves analyzing the values of other variables for a given value of the conditioned variable. In the first example, conditioning on B implies that observations for a given value of B should show no correlation between A and C. If such a correlation exists, then the model is incorrect.
What is the value of a random variable?
The expected value of a random variable is denoted by E[X]. The expected value can be thought of as the “average” value attained by the random variable; in fact, the expected value of a random variable is also called its mean, in which case we use the notation µX. (µ is the Greek letter mu.)
How many values can a random variable take?
A discrete random variable can take only a finite number of distinct values such as 0, 1, 2, 3, 4, … and so on. The probability distribution of a random variable has a list of probabilities compared with each of its possible values known as probability mass function.
How are random variables treated in probability theory?
The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers.
When is a random variable said to be continuous?
If the random variable X can assume an infinite and uncountable set of values, it is said to be a continuous random variable. When X takes any value in a given interval (a, b), it is said to be a continuous random variable in that interval. Formally, a continuous random variable is such whose cumulative distribution function is constant throughout.
How is a random variable defined as a measurable function?
In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers. This graph shows how random variable is a function from all possible outcomes to real values.