Find the generating functions of
a)p{x<=n+1}
b)p{x=2n}
To find the generating functions for the given probabilities, we need to understand what "generating functions" mean in this context.
A generating function is a power series representation of a sequence or a set of numbers. In the context of probability, the generating function is used to encode the probabilities of different outcomes into a single function.
Let's break down the given probabilities and find their generating functions.
a) p{x <= n+1}:
This probability represents the sum of probabilities of all favorable outcomes where x is less than or equal to n+1.
To find the generating function, we need to represent the probability of each outcome as a coefficient in a power series.
Let's consider each term in the power series representation separately:
- The coefficient for the term x^0 represents the probability of x = 0, which is p{x = 0}.
- The coefficient for the term x^1 represents the probability of x = 1, which is p{x = 1}.
- ...
- The coefficient for the term x^n represents the probability of x = n, which is p{x = n}.
- The coefficient for the term x^(n+1) represents the probability of x = n+1, which is p{x = n+1}.
To get the generating function, we can write:
G_a(x) = p{x = 0} * x^0 + p{x = 1} * x^1 + ... + p{x = n} * x^n + p{x = n+1} * x^(n+1)
b) p{x = 2n}:
This probability represents the probability of a specific event where x equals 2n.
To find the generating function, we need to represent this probability as a coefficient in a power series.
Let's consider each term in the power series representation:
- The coefficient for the term x^(2n) represents the probability of x = 2n, which is p{x = 2n}.
To get the generating function, we can write:
G_b(x) = p{x = 2n} * x^(2n)
These generating functions allow us to encode the probabilities into a single mathematical function, which can be used for various calculations and analysis.