The Probability Distribution Of A Discrete Random Variable $X$ Is Given Below.$\[ \begin{tabular}{|c|r|r|r|r|r|r|} \hline $x$ & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline $P(x)$ & 0.6 & 0.17 & 0.04 & 0.06 & 0.11 & 0.02

The Probability Distribution of a Discrete Random Variable: Understanding the Concept

In probability theory, a discrete random variable is a variable that can take on a countable number of distinct values. The probability distribution of a discrete random variable is a function that assigns a probability to each possible value of the variable. In this article, we will discuss the probability distribution of a discrete random variable and provide a step-by-step guide on how to calculate the probability of each value.

What is a Discrete Random Variable?

A discrete random variable is a variable that can take on a countable number of distinct values. For example, the number of heads obtained when flipping a coin three times is a discrete random variable, as it can take on the values 0, 1, 2, or 3. Similarly, the number of students in a class who have a certain grade is a discrete random variable, as it can take on the values 0, 1, 2, and so on.

The Probability Distribution of a Discrete Random Variable

The probability distribution of a discrete random variable is a function that assigns a probability to each possible value of the variable. The probability distribution is typically represented as a table or a graph, where the x-axis represents the values of the variable and the y-axis represents the corresponding probabilities.

Example: Probability Distribution of a Discrete Random Variable

Let's consider the probability distribution of a discrete random variable X, which is given below:

In this example, the variable X can take on the values 1, 2, 3, 4, 5, and 6, and the corresponding probabilities are 0.6, 0.17, 0.04, 0.06, 0.11, and 0.02, respectively.

Calculating the Probability of Each Value

To calculate the probability of each value, we need to multiply the probability of each value by the number of times it occurs. For example, the probability of the value 1 is 0.6, and it occurs 1 time, so the probability of the value 1 is 0.6. Similarly, the probability of the value 2 is 0.17, and it occurs 1 time, so the probability of the value 2 is 0.17.

Calculating the Expected Value

The expected value of a discrete random variable is the sum of the product of each value and its probability. To calculate the expected value, we need to multiply each value by its probability and sum the results. For example, the expected value of the variable X is:

E(X) = (1 x 0.6) + (2 x 0.17) + (3 x 0.04) + (4 x 0.06) + (5 x 0.11) + (6 x 0.02) = 0.6 + 0.34 + 0.12 + 0.24 + 0.55 + 0.12 = 1.97

Calculating the Variance

The variance of a discrete random variable is the sum of the product of each value and its squared probability. To calculate the variance, we need to multiply each value by its squared probability and sum the results. For example, the variance of the variable X is:

Var(X) = (1 x 0.6^2) + (2 x 0.17^2) + (3 x 0.04^2) + (4 x 0.06^2) + (5 x 0.11^2) + (6 x 0.02^2) = 0.36 + 0.0289 + 0.0016 + 0.0072 + 0.0605 + 0.0004 = 0.4346

In this article, we discussed the probability distribution of a discrete random variable and provided a step-by-step guide on how to calculate the probability of each value. We also calculated the expected value and variance of the variable X using the given probability distribution. The probability distribution of a discrete random variable is an important concept in probability theory, and it has many applications in statistics, engineering, and other fields.

• [1] Ross, S. M. (2010). A First Course in Probability. 8th ed. Pearson Education.
• [2] Johnson, R. A. (2013). Probability and Statistics for Engineers. 2nd ed. Cengage Learning.
• [3] Papoulis, A. (2002). Probability, Random Variables, and Stochastic Processes. 4th ed. McGraw-Hill.

• Discrete random variable: A variable that can take on a countable number of distinct values.
• Probability distribution: A function that assigns a probability to each possible value of a discrete random variable.
• Expected value: The sum of the product of each value and its probability.
• Variance: The sum of the product of each value and its squared probability.
  Frequently Asked Questions (FAQs) about Discrete Random Variables

Q: What is a discrete random variable?

A: A discrete random variable is a variable that can take on a countable number of distinct values. For example, the number of heads obtained when flipping a coin three times is a discrete random variable, as it can take on the values 0, 1, 2, or 3.

Q: What is the probability distribution of a discrete random variable?

A: The probability distribution of a discrete random variable is a function that assigns a probability to each possible value of the variable. The probability distribution is typically represented as a table or a graph, where the x-axis represents the values of the variable and the y-axis represents the corresponding probabilities.

Q: How do I calculate the probability of each value in a discrete random variable?

A: To calculate the probability of each value, you need to multiply the probability of each value by the number of times it occurs. For example, if the probability of the value 1 is 0.6 and it occurs 1 time, the probability of the value 1 is 0.6.

Q: What is the expected value of a discrete random variable?

A: The expected value of a discrete random variable is the sum of the product of each value and its probability. To calculate the expected value, you need to multiply each value by its probability and sum the results.

Q: What is the variance of a discrete random variable?

A: The variance of a discrete random variable is the sum of the product of each value and its squared probability. To calculate the variance, you need to multiply each value by its squared probability and sum the results.

Q: How do I use the probability distribution of a discrete random variable in real-life applications?

A: The probability distribution of a discrete random variable has many applications in statistics, engineering, and other fields. For example, it can be used to model the number of defects in a manufacturing process, the number of customers in a store, or the number of accidents on a road.

Q: What are some common types of discrete random variables?

A: Some common types of discrete random variables include:

• Bernoulli random variable: A random variable that can take on the values 0 or 1.
• Binomial random variable: A random variable that can take on the values 0, 1, 2, ..., n, where n is a positive integer.
• Poisson random variable: A random variable that can take on the values 0, 1, 2, ..., where the probability of each value is proportional to the value.

Q: How do I determine the type of discrete random variable I have?

A: To determine the type of discrete random variable you have, you need to examine the probability distribution of the variable. For example, if the probability distribution is a table with two columns (x and P(x)), and the values in the x column are 0 and 1, then the variable is a Bernoulli random variable.

Q: What are some common applications of discrete random variables?

A: Some common applications of discrete random variables include:

• Quality control: Discrete random variables can be used to model the number of defects in a manufacturing process.
• Finance: Discrete random variables can be used to model the number of customers in a store or the number of accidents on a road.
• Engineering: Discrete random variables can be used to model the number of components in a system or the number of failures in a system.

Q: How do I calculate the probability of a discrete random variable using a calculator or computer?

A: To calculate the probability of a discrete random variable using a calculator or computer, you can use a statistical software package such as R or Python. These packages have built-in functions for calculating the probability of a discrete random variable.

Q: What are some common mistakes to avoid when working with discrete random variables?

A: Some common mistakes to avoid when working with discrete random variables include:

• Not checking the probability distribution: Make sure to check the probability distribution of the variable to ensure that it is correct.
• Not using the correct formula: Make sure to use the correct formula for calculating the expected value and variance of the variable.
• Not considering the type of discrete random variable: Make sure to consider the type of discrete random variable you have and use the correct formula for calculating the expected value and variance.