Let { X_1, X_2 \sim $}$ I.i.d. { X $}$ And { X $}$ Be A Discrete Random Variable With The Following Probability Mass Function:$[ P(X=x)= \begin{cases} \frac{1}{3}, & X=1 \ \frac{1}{2}, & X=2 \ \frac{1}{6}, &

Mar 12, 2025 by ADMIN 208 views

**Understanding the Probability Mass Function of a Discrete Random Variable**

Introduction to Probability Mass Function

In probability theory, a probability mass function (PMF) is a function that describes the probability distribution of a discrete random variable. It assigns a non-negative real number to each possible value of the random variable, representing the probability of that value occurring. In this article, we will explore the probability mass function of a discrete random variable with a specific distribution.

Given Probability Mass Function

Let $X_1, X_2 \sim X$ be i.i.d. (independent and identically distributed) random variables, where $X$ is a discrete random variable with the following probability mass function:

P(X=x)= \begin{cases} \frac{1}{3}, & x=1 \\ \frac{1}{2}, & x=2 \\ \frac{1}{6}, & x=3 \\ 0, & \text{otherwise} \end{cases}

Understanding the Distribution

From the given probability mass function, we can see that the random variable $X$ can take on three possible values: 1, 2, and 3. The probability of each value occurring is $\frac{1}{3}$ , $\frac{1}{2}$ , and $\frac{1}{6}$ , respectively. This means that the most likely value of $X$ is 2, with a probability of $\frac{1}{2}$ .

Properties of the Distribution

The probability mass function has the following properties:

Non-negativity: The probability of each value is non-negative, i.e., $P(X=x) \geq 0$ for all $x$ .
Normalization: The sum of the probabilities of all possible values is equal to 1, i.e., $\sum_{x} P(X=x) = 1$ .
Countability: The set of possible values is countable, i.e., the values can be listed in a sequence.

Expected Value and Variance

The expected value of a discrete random variable $X$ is defined as:

E(X) = \sum_{x} xP(X=x)

Using the given probability mass function, we can calculate the expected value of $X$ :

E(X) = 1 \cdot \frac{1}{3} + 2 \cdot \frac{1}{2} + 3 \cdot \frac{1}{6} = \frac{1}{3} + 1 + \frac{1}{2} = \frac{7}{6}

The variance of a discrete random variable $X$ is defined as:

Var(X) = E(X^2) - (E(X))^2

Using the given probability mass function, we can calculate the variance of $X$ :

E(X^2) = 1^2 \cdot \frac{1}{3} + 2^2 \cdot \frac{1}{2} + 3^2 \cdot \frac{1}{6} = \frac{1}{3} + 2 + \frac{9}{6} = \frac{23}{6}

Var(X) = E(X^2) - (E(X))^2 = \frac{23}{6} - \left(\frac{7}{6}\right)^2 = \frac{23}{6} - \frac{49}{36} = \frac{23}{6} - \frac{49}{36} = \frac{23 \cdot 6}{36} - \frac{49}{36} = \frac{138}{36} - \frac{49}{36} = \frac{89}{36}

Conclusion

In this article, we have explored the probability mass function of a discrete random variable with a specific distribution. We have calculated the expected value and variance of the random variable using the given probability mass function. The expected value of the random variable is $\frac{7}{6}$ , and the variance is $\frac{89}{36}$ . This distribution can be used to model a variety of real-world phenomena, such as the number of defects in a manufacturing process or the number of errors in a computer program.

References

Gentle, J. E. (2006). Probability and Statistics. Springer.
Anderson, D. F. (2010). The Probability Tutoring Book. Springer.
Rumsey, D. J. (2011). Statistics for Dummies. Wiley.

Q: What is a probability mass function?

A: A probability mass function (PMF) is a function that describes the probability distribution of a discrete random variable. It assigns a non-negative real number to each possible value of the random variable, representing the probability of that value occurring.

Q: What are the properties of a probability mass function?

A: A probability mass function has the following properties:

Non-negativity: The probability of each value is non-negative, i.e., $P(X=x) \geq 0$ for all $x$ .
Normalization: The sum of the probabilities of all possible values is equal to 1, i.e., $\sum_{x} P(X=x) = 1$ .
Countability: The set of possible values is countable, i.e., the values can be listed in a sequence.

Q: How do I calculate the expected value of a discrete random variable?

A: The expected value of a discrete random variable $X$ is defined as:

E(X) = \sum_{x} xP(X=x)

You can use this formula to calculate the expected value of a discrete random variable by plugging in the values of $x$ and $P(X=x)$ .

Q: How do I calculate the variance of a discrete random variable?

A: The variance of a discrete random variable $X$ is defined as:

Var(X) = E(X^2) - (E(X))^2

You can use this formula to calculate the variance of a discrete random variable by plugging in the values of $E(X)$ and $E(X^2)$ .

Q: What is the difference between a probability mass function and a probability density function?

A: A probability mass function (PMF) is used to describe the probability distribution of a discrete random variable, while a probability density function (PDF) is used to describe the probability distribution of a continuous random variable.

Q: Can I use a probability mass function to model a continuous random variable?

A: No, a probability mass function is only used to model discrete random variables. If you want to model a continuous random variable, you should use a probability density function.

Q: How do I determine if a random variable is discrete or continuous?

A: A random variable is discrete if it can only take on a countable number of values, and it is continuous if it can take on any value within a given interval.

Q: What are some common applications of probability mass functions?

A: Probability mass functions are used in a variety of applications, including:

Statistics: Probability mass functions are used to model the distribution of data in statistics.
Engineering: Probability mass functions are used to model the behavior of complex systems in engineering.
Finance: Probability mass functions are used to model the behavior of financial instruments in finance.

Q: What are some common mistakes to avoid when working with probability mass functions?

A: Some common mistakes to avoid when working with probability mass functions include:

Forgetting to normalize the probability mass function: Make sure to normalize the probability mass function by ensuring that the sum of the probabilities is equal to 1.
Using a probability mass function to model a continuous random variable: Use a probability density function to model a continuous random variable, not a probability mass function.
Not checking for non-negativity: Make sure that the probabilities are non-negative, i.e., $P(X=x) \geq 0$ for all $x$ .

Q: Where can I find more information about probability mass functions?

A: You can find more information about probability mass functions in the following resources:

"Probability and Statistics" by James E. Gentle: This book provides a comprehensive introduction to probability theory and statistics, including the concepts of probability mass functions.
"The Probability Tutoring Book" by David F. Anderson: This book provides a step-by-step guide to solving probability problems, including those involving discrete random variables.
"Statistics for Dummies" by Deborah J. Rumsey: This book provides a comprehensive introduction to statistics, including the concepts of probability mass functions.