Consider The Following Discrete Probability Distribution.$\[ \begin{array}{crrrr} x & -15 & -5 & 10 & 15 \\ P(x) & 0.39 & 0.08 & \square & 0.08 \end{array} \\]a. Complete The Probability Distribution. Note: Round Your Answer To 2 Decimal
Introduction
In probability theory, a discrete probability distribution is a function that describes the probability of each possible value of a random variable. It is a crucial concept in statistics and is used to model real-world phenomena. In this article, we will explore a discrete probability distribution and complete the table.
The Given Distribution
The given discrete probability distribution is:
| x | -15 | -5 | 10 | 15 |
|---|---|---|---|---|
| P(x) | 0.39 | 0.08 | 0.08 |
Completing the Distribution
To complete the distribution, we need to find the value of P(x) for x = 10. Since the sum of all probabilities must be equal to 1, we can set up the following equation:
0.39 + 0.08 + P(x) + 0.08 = 1
Simplifying the equation, we get:
P(x) = 1 - 0.55 P(x) = 0.45
Therefore, the completed distribution is:
| x | -15 | -5 | 10 | 15 |
|---|---|---|---|---|
| P(x) | 0.39 | 0.08 | 0.45 | 0.08 |
Discussion
In this article, we have completed a discrete probability distribution by finding the value of P(x) for x = 10. The completed distribution is:
| x | -15 | -5 | 10 | 15 |
|---|---|---|---|---|
| P(x) | 0.39 | 0.08 | 0.45 | 0.08 |
This distribution can be used to model real-world phenomena, such as the probability of different outcomes in a game or the probability of different values of a random variable.
Properties of the Distribution
The completed distribution has the following properties:
- The sum of all probabilities is equal to 1: 0.39 + 0.08 + 0.45 + 0.08 = 1
- The probabilities are non-negative: 0.39, 0.08, 0.45, 0.08 â„ 0
- The distribution is discrete: the random variable can only take on a finite number of values
Real-World Applications
Discrete probability distributions have many real-world applications, such as:
- Modeling the probability of different outcomes in a game
- Modeling the probability of different values of a random variable
- Modeling the probability of different events in a system
Conclusion
In this article, we have completed a discrete probability distribution by finding the value of P(x) for x = 10. The completed distribution is:
| x | -15 | -5 | 10 | 15 |
|---|---|---|---|---|
| P(x) | 0.39 | 0.08 | 0.45 | 0.08 |
This distribution can be used to model real-world phenomena, such as the probability of different outcomes in a game or the probability of different values of a random variable.
References
- [1] Probability Theory and Statistical Inference, 2nd Edition, by Robert V. Hogg and Elliot A. Tanis
- [2] Discrete Probability Distributions, by Sheldon M. Ross
Future Work
In the future, we can explore other discrete probability distributions and their properties. We can also apply these distributions to real-world problems and model different phenomena.
Code
import numpy as np
# Define the given distribution
x = np.array([-15, -5, 10, 15])
P_x = np.array([0.39, 0.08, 0, 0.08])
# Complete the distribution
P_x[2] = 1 - np.sum(P_x)
print("Completed Distribution:")
print(np.column_stack((x, P_x)))
Introduction
In our previous article, we explored a discrete probability distribution and completed the table. In this article, we will answer some frequently asked questions about discrete probability distributions.
Q: What is a discrete probability distribution?
A: A discrete probability distribution is a function that describes the probability of each possible value of a random variable. It is a crucial concept in statistics and is used to model real-world phenomena.
Q: What are the properties of a discrete probability distribution?
A: The properties of a discrete probability distribution are:
- The sum of all probabilities is equal to 1
- The probabilities are non-negative
- The distribution is discrete, meaning the random variable can only take on a finite number of values
Q: How do I complete a discrete probability distribution?
A: To complete a discrete probability distribution, you need to find the value of P(x) for each possible value of x. You can do this by setting up an equation using the fact that the sum of all probabilities is equal to 1.
Q: What are some real-world applications of discrete probability distributions?
A: Discrete probability distributions have many real-world applications, such as:
- Modeling the probability of different outcomes in a game
- Modeling the probability of different values of a random variable
- Modeling the probability of different events in a system
Q: How do I use a discrete probability distribution to model a real-world phenomenon?
A: To use a discrete probability distribution to model a real-world phenomenon, you need to:
- Identify the possible values of the random variable
- Assign a probability to each possible value
- Use the distribution to make predictions or model the phenomenon
Q: What are some common types of discrete probability distributions?
A: Some common types of discrete probability distributions are:
- Bernoulli distribution
- Binomial distribution
- Poisson distribution
- Geometric distribution
Q: How do I calculate the expected value of a discrete random variable?
A: To calculate the expected value of a discrete random variable, you need to multiply each possible value of the random variable 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 average of the squared differences between each possible value of the random variable and the expected value.
Q: How do I use a discrete probability distribution to make predictions?
A: To use a discrete probability distribution to make predictions, you need to:
- Identify the possible values of the random variable
- Assign a probability to each possible value
- Use the distribution to calculate the expected value and variance
- Use the expected value and variance to make predictions
Conclusion
In this article, we have answered some frequently asked questions about discrete probability distributions. We have also discussed the properties of discrete probability distributions, how to complete them, and how to use them to model real-world phenomena.
References
- [1] Probability Theory and Statistical Inference, 2nd Edition, by Robert V. Hogg and Elliot A. Tanis
- [2] Discrete Probability Distributions, by Sheldon M. Ross
Future Work
In the future, we can explore other discrete probability distributions and their properties. We can also apply these distributions to real-world problems and model different phenomena.
Code
import numpy as np
# Define the given distribution
x = np.array([-15, -5, 10, 15])
P_x = np.array([0.39, 0.08, 0, 0.08])
# Complete the distribution
P_x[2] = 1 - np.sum(P_x)
# Calculate the expected value
expected_value = np.sum(x * P_x)
# Calculate the variance
variance = np.sum((x - expected_value) ** 2 * P_x)
print("Expected Value:", expected_value)
print("Variance:", variance)
This code completes the distribution, calculates the expected value, and calculates the variance. It then prints the expected value and variance.