Finding the probability distribution of a random variable

Oct 13, 2025 by windailys.com 26 views

Let $Y_1, Y_2, Y_3, \ldots, Y_n$ be $n$ independent Bernoulli trials with probability $p$ . Let $S_n = Y_1 + Y_2 + Y_3 + \ldots + Y_n$ . Find the probability generating function of $S_n$

In probability theory, the probability generating function (PGF) is a powerful tool used to study the properties of discrete random variables. Given a sequence of independent Bernoulli trials, we can use the PGF to find the probability distribution of the sum of these trials. In this article, we will derive the probability generating function of $S_n$ , the sum of $n$ independent Bernoulli trials with probability $p$ .

A Bernoulli trial is a random experiment with two possible outcomes: success or failure. The probability of success is denoted by $p$ , and the probability of failure is $1-p$ . In this case, we have $n$ independent Bernoulli trials, denoted by $Y_1, Y_2, Y_3, \ldots, Y_n$ . Each trial is independent of the others, and the probability of success for each trial is $p$ .

The probability generating function (PGF) of a discrete random variable $X$ is defined as:

G_X(z) = E(z^X) = \sum_{x} z^x P(X=x)

where $x$ takes on all possible values of $X$ . For a Bernoulli trial, the PGF is given by:

G_Y(z) = E(z^Y) = p z + (1-p)

To find the PGF of $S_n$ , we can use the fact that the sum of independent random variables is also independent. Therefore, we can write:

G_{S_n}(z) = E(z^{S_n}) = E(z^{Y_1 + Y_2 + \ldots + Y_n})

Using the property of the PGF, we can rewrite this as:

G_{S_n}(z) = E(z^{Y_1}) E(z^{Y_2}) \ldots E(z^{Y_n})

Since each trial is independent, we can simplify this to:

G_{S_n}(z) = (p z + (1-p)) (p z + (1-p)) \ldots (p z + (1-p))

where the product is taken $n$ times. This can be rewritten as:

G_{S_n}(z) = (p z + (1-p))^n

The PGF of $S_n$ has several important properties. First, it is a polynomial of degree $n$ in $z$ . Second, it is a symmetric function of the $n$ trials, meaning that the order of the trials does not matter. Finally, the PGF is a continuous function of $z$ , meaning that small changes in $z$ result in small changes in the PGF.

The PGF can be used to find the moments of $S_n$ . The $k$ th moment of $S_n$ is given by:

E(S_n^k) = \left. \frac{d^k}{dz^k} G_{S_n}(z) \right|_{z=1}

Using the PGF, we can find the first few moments of $S_n$ :

E(S_n) = np

E(S_n^2) = n p (1-p) + n^2 p^2

E(S_n^3) = n p (1-p)^2 + 3 n^2 p^2 (1-p) + n^3 p^3

In this article, we derived the probability generating function of $S_n$ , the sum of $n$ independent Bernoulli trials with probability $p$ . We showed that the PGF is a polynomial of degree $n$ in $z$ , and that it is a symmetric function of the $n$ trials. We also used the PGF to find the moments of $S_n$ . The PGF is a powerful tool for studying the properties of discrete random variables, and it has many applications in probability theory and statistics.

Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. 1. John Wiley & Sons.
Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and Random Processes. Oxford University Press.
Ross, S. M. (2014). Introduction to Probability Models. Academic Press.

For a more detailed treatment of the probability generating function, see Feller (1968) or Grimmett & Stirzaker (2001).
For a more advanced treatment of probability theory, see Ross (2014).
For applications of the probability generating function in statistics, see the references listed in the references section.
Q&A: Probability Generating Function of $S_n$ =============================================

Q: What is the probability generating function of $S_n$ ?

A: The probability generating function of $S_n$ is given by:

G_{S_n}(z) = (p z + (1-p))^n

Q: What are the properties of the probability generating function of $S_n$ ?

A: The probability generating function of $S_n$ has several important properties. First, it is a polynomial of degree $n$ in $z$ . Second, it is a symmetric function of the $n$ trials, meaning that the order of the trials does not matter. Finally, the PGF is a continuous function of $z$ , meaning that small changes in $z$ result in small changes in the PGF.

Q: How can I use the probability generating function of $S_n$ to find the moments of $S_n$ ?

A: The $k$ th moment of $S_n$ is given by:

E(S_n^k) = \left. \frac{d^k}{dz^k} G_{S_n}(z) \right|_{z=1}

Using the PGF, we can find the first few moments of $S_n$ :

E(S_n) = np

E(S_n^2) = n p (1-p) + n^2 p^2

E(S_n^3) = n p (1-p)^2 + 3 n^2 p^2 (1-p) + n^3 p^3

Q: What are some applications of the probability generating function of $S_n$ ?

A: The probability generating function of $S_n$ has many applications in probability theory and statistics. Some examples include:

Random walks: The PGF can be used to study the properties of random walks, which are sequences of independent and identically distributed random variables.
Markov chains: The PGF can be used to study the properties of Markov chains, which are sequences of random variables that satisfy certain probabilistic constraints.
Queueing theory: The PGF can be used to study the properties of queueing systems, which are systems that consist of a queue of customers waiting to be served.

Q: How can I use the probability generating function of $S_n$ to solve problems in probability theory and statistics?

A: The probability generating function of $S_n$ can be used to solve a wide range of problems in probability theory and statistics. Some examples include:

Finding the probability distribution of a random variable: The PGF can be used to find the probability distribution of a random variable, which is a function that assigns a probability to each possible value of the variable.
Finding the expected value of a random variable: The PGF can be used to find the expected value of a random variable, which is a measure of the central tendency of the variable.
Finding the variance of a random variable: The PGF can be used to find the variance of a random variable, which is a measure of the spread of the variable.

Q: What are some common mistakes to avoid when using the probability generating function of $S_n$ ?

A: Some common mistakes to avoid when using the probability generating function of $S_n$ include:

Not checking the convergence of the PGF: The PGF must be convergent for the results to be valid.
Not checking the symmetry of the PGF: The PGF must be symmetric for the results to be valid.
Not checking the continuity of the PGF: The PGF must be continuous for the results to be valid.

Q: Where can I find more information about the probability generating function of $S_n$ ?

A: There are many resources available for learning more about the probability generating function of $S_n$ . Some examples include:

Textbooks: There are many textbooks that cover the probability generating function of $S_n$ , including Feller (1968) and Grimmett & Stirzaker (2001).
Online resources: There are many online resources available for learning more about the probability generating function of $S_n$ , including Wikipedia and MathWorld.
Research papers: There are many research papers that cover the probability generating function of $S_n$ , including papers on random walks and Markov chains.