Let X be a random variable that takes integer values, with PMF pX(x) . Let Y be another integer-valued random variable and let y be a number.
a) Is pX(y) a random variable or a number?
b) Is pX(Y) a random variable or a number?
number
random variable
a) Well, pX(y) is just the probability of X taking on the value y, so it's actually a number. It's like finding out how many clowns can fit into a tiny car - it's a specific answer, not a random variable.
b) On the other hand, pX(Y) is a random variable because Y is a random variable. This is like asking how many clowns can fit into a car that itself can hold a random number of clowns. The number of clowns that can fit will depend on the outcome of Y, making it a random variable. The joys of clown car physics!
a) The notation pX(y) represents the probability mass function (PMF) of the random variable X evaluated at the value y. Therefore, pX(y) is a number, specifically the probability of the event X = y.
b) The notation pX(Y) represents the probability mass function (PMF) of the random variable X evaluated at the random variable Y. Since Y is also a random variable, pX(Y) is itself a random variable. It represents the distribution of X when the value of Y is used as an input to the PMF pX(x).
To answer these questions, we need to understand the definitions of random variables.
a) The expression pX(y) represents the probability mass function (PMF) of the random variable X evaluated at the value y. In this case, pX(y) is a number, not a random variable. The PMF gives the probabilities associated with each value of the random variable X.
b) On the other hand, pX(Y) represents the PMF of the random variable X evaluated at the random variable Y. In this case, pX(Y) is a random variable. The random variable Y is plugged into the PMF of X, and the resulting expression is a new random variable.
In summary:
a) pX(y) is a number.
b) pX(Y) is a random variable.