The Following Table Shows The Probability Distribution For A Discrete Random Variable.$\[ \begin{tabular}{|c|c|c|c|c|c|c|} \hline $X$ & 24 & 26 & 27 & 32 & 35 & 39 \\ \hline $P(X)$ & 0.16 & 0.09 & 0.18 & 0.12 & 0.24 & 0.21

Introduction

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 analyze the probability distribution of a discrete random variable, using the following table as a reference.

The Probability Distribution Table

$X$	24	26	27	32	35	39
$P(X)$	0.16	0.09	0.18	0.12	0.24	0.21

Understanding the Probability Distribution

The probability distribution of a discrete random variable is a function that assigns a probability to each possible value of the variable. In this case, the probability distribution is given by the table above, where each value of $X$ is associated with a probability $P(X)$. The probability distribution is a way of describing the uncertainty of the variable, and it is used to calculate the probability of different events.

Properties of the Probability Distribution

The probability distribution of a discrete random variable has several important properties. These properties are:

• Non-negativity: The probability of each value of the variable is non-negative, i.e., $P(X) \geq 0$ for all $X$.
• Normalization: The sum of the probabilities of all possible values of the variable is equal to 1, i.e., $\sum_{X} P(X) = 1$.
• Countable number of values: The variable can take on a countable number of distinct values.

Calculating Probabilities

The probability distribution of a discrete random variable can be used to calculate the probability of different events. For example, the probability of the variable taking on a value greater than 30 can be calculated as follows:

$$P(X > 30) = P(32) + P(35) + P(39) = 0.12 + 0.24 + 0.21 = 0.57$$

Similarly, the probability of the variable taking on a value less than 27 can be calculated as follows:

$$P(X < 27) = P(24) + P(26) = 0.16 + 0.09 = 0.25$$

Expected Value

The expected value of a discrete random variable is a measure of the central tendency of the variable. It is calculated as follows:

$$E(X) = \sum_{X} X \cdot P(X)$$

In this case, the expected value of the variable is:

$$E(X) = 24 \cdot 0.16 + 26 \cdot 0.09 + 27 \cdot 0.18 + 32 \cdot 0.12 + 35 \cdot 0.24 + 39 \cdot 0.21 = 29.32$$

Variance

The variance of a discrete random variable is a measure of the spread of the variable. It is calculated as follows:

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

In this case, the variance of the variable is:

$$Var(X) = (24^2 \cdot 0.16 + 26^2 \cdot 0.09 + 27^2 \cdot 0.18 + 32^2 \cdot 0.12 + 35^2 \cdot 0.24 + 39^2 \cdot 0.21) - (29.32)^2 = 12.45$$

Conclusion

In this article, we have analyzed the probability distribution of a discrete random variable, using the following table as a reference. We have discussed the properties of the probability distribution, calculated probabilities, and calculated the expected value and variance of the variable. The probability distribution of a discrete random variable is a powerful tool for describing the uncertainty of the variable, and it is used to calculate the probability of different events.

References

• [1] Ross, S. M. (2010). A First Course in Probability. 8th ed. Pearson Education.
• [2] Sheldon M. Ross (2012). Introduction to Probability Models. 10th ed. Academic Press.

Further Reading

• Discrete Random Variables: A discrete random variable is a variable that can take on a countable number of distinct values.
• Probability Distribution: The probability distribution of a discrete random variable is a function that assigns a probability to each possible value of the variable.
• Expected Value: The expected value of a discrete random variable is a measure of the central tendency of the variable.
• Variance: The variance of a discrete random variable is a measure of the spread of the variable.
  Frequently Asked Questions (FAQs) about the Probability Distribution of a Discrete Random Variable =============================================================================================

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. In other words, it is a variable that can take on a finite or countably infinite number of values.

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. It is a way of describing the uncertainty of the variable, and it is used to calculate the probability of different events.

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

A: The probability distribution of a discrete random variable has several important properties, including:

• Non-negativity: The probability of each value of the variable is non-negative, i.e., $P(X) \geq 0$ for all $X$.
• Normalization: The sum of the probabilities of all possible values of the variable is equal to 1, i.e., $\sum_{X} P(X) = 1$.
• Countable number of values: The variable can take on a countable number of distinct values.

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

A: To calculate the probability of a discrete random variable, you need to use the probability distribution of the variable. For example, if you want to calculate the probability of the variable taking on a value greater than 30, you would use the following formula:

$$P(X > 30) = P(32) + P(35) + P(39) = 0.12 + 0.24 + 0.21 = 0.57$$

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

A: The expected value of a discrete random variable is a measure of the central tendency of the variable. It is calculated as follows:

$$E(X) = \sum_{X} X \cdot P(X)$$

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

A: The variance of a discrete random variable is a measure of the spread of the variable. It is calculated as follows:

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

Q: How do I use the probability distribution of a discrete random variable to make decisions?

A: The probability distribution of a discrete random variable can be used to make decisions by calculating the probability of different events. For example, if you want to decide whether to invest in a particular stock, you can use the probability distribution of the stock's price to calculate the probability of the stock's price increasing or decreasing.

Q: What are some common applications of the probability distribution of a discrete random variable?

A: The probability distribution of a discrete random variable has many applications in fields such as finance, engineering, and economics. Some common applications include:

• Risk analysis: The probability distribution of a discrete random variable can be used to calculate the probability of different risks, such as the probability of a stock's price increasing or decreasing.
• Decision-making: The probability distribution of a discrete random variable can be used to make decisions by calculating the probability of different events.
• Optimization: The probability distribution of a discrete random variable can be used to optimize systems by calculating the probability of different outcomes.

Q: What are some common mistakes to avoid when working with the probability distribution of a discrete random variable?

A: Some common mistakes to avoid when working with the probability distribution of a discrete random variable include:

• Not normalizing the probability distribution: The probability distribution of a discrete random variable must be normalized, i.e., the sum of the probabilities of all possible values of the variable must be equal to 1.
• Not using the correct formula for the expected value and variance: The expected value and variance of a discrete random variable are calculated using the following formulas:

$$E(X) = \sum_{X} X \cdot P(X)$$

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

Q: What are some common tools and techniques used to work with the probability distribution of a discrete random variable?

A: Some common tools and techniques used to work with the probability distribution of a discrete random variable include:

• Probability tables: A probability table is a table that lists the possible values of a discrete random variable and their corresponding probabilities.
• Probability distributions: A probability distribution is a function that assigns a probability to each possible value of a discrete random variable.
• Expected value and variance calculations: The expected value and variance of a discrete random variable can be calculated using the following formulas:

$$E(X) = \sum_{X} X \cdot P(X)$$

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

Conclusion

In this article, we have answered some frequently asked questions about the probability distribution of a discrete random variable. We have discussed the properties of the probability distribution, how to calculate the probability of a discrete random variable, and how to use the probability distribution to make decisions. We have also discussed some common applications of the probability distribution of a discrete random variable and some common mistakes to avoid when working with it.