Consider The Following Discrete Probability Distribution.${ \begin{array}{|c|c|c|c|c|} \hline x & -25 & -15 & 10 & 20 \ \hline \text{Probability} & 0.35 & 0.10 & & 0.10 \ \hline \end{array} }$a. Complete The Probability Distribution.b.

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Introduction

In probability theory, a discrete probability distribution is a function that describes the probability of each possible outcome of a random variable. It is a crucial concept in statistics and is used to model real-world phenomena. In this article, we will consider a given discrete probability distribution and complete the table.

The Given Discrete Probability Distribution

The given discrete probability distribution is as follows:

x -25 -15 10 20
Probability 0.35 0.10 0.10

Completing the Probability Distribution

To complete the probability distribution, we need to find the missing probabilities for the outcomes x = 10 and x = 20. We know that the sum of all probabilities in a discrete probability distribution must be equal to 1.

Let's denote the missing probability for x = 10 as P(10) and the missing probability for x = 20 as P(20). We can write the following equation:

P(-25) + P(-15) + P(10) + P(20) = 1

Substituting the given probabilities, we get:

0.35 + 0.10 + P(10) + P(20) = 1

Simplifying the equation, we get:

0.45 + P(10) + P(20) = 1

Subtracting 0.45 from both sides, we get:

P(10) + P(20) = 0.55

Since the probabilities must be non-negative, we can divide the equation by 2 to get:

P(10) = P(20) = 0.275

Therefore, the completed probability distribution is as follows:

x -25 -15 10 20
Probability 0.35 0.10 0.275 0.275

Discussion

In this article, we considered a given discrete probability distribution and completed the table. We used the fact that the sum of all probabilities in a discrete probability distribution must be equal to 1 to find the missing probabilities. The completed probability distribution is a useful tool for modeling real-world phenomena and making predictions.

Properties of Discrete Probability Distributions

Discrete probability distributions have several important properties that make them useful for modeling real-world phenomena. Some of these properties include:

  • Non-negativity: The probability of each outcome must be non-negative.
  • Normalization: The sum of all probabilities must be equal to 1.
  • Countable outcomes: The number of possible outcomes must be countable.
  • Probability axioms: The probability distribution must satisfy the probability axioms, which include the axioms of non-negativity, normalization, and countable additivity.

Examples of Discrete Probability Distributions

Discrete probability distributions are used to model a wide range of real-world phenomena, including:

  • Coin tossing: The probability distribution of the number of heads in a sequence of coin tosses.
  • Rolling a die: The probability distribution of the number of dots on a die.
  • Random sampling: The probability distribution of the number of successes in a random sample.

Conclusion

In this article, we considered a given discrete probability distribution and completed the table. We used the fact that the sum of all probabilities in a discrete probability distribution must be equal to 1 to find the missing probabilities. The completed probability distribution is a useful tool for modeling real-world phenomena and making predictions. We also discussed the properties of discrete probability distributions and provided examples of their use in modeling real-world phenomena.

References

  • Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer.
  • Feller, W. (1950). An Introduction to Probability Theory and Its Applications. Wiley.
  • Ross, S. M. (2010). A First Course in Probability. Prentice Hall.

Further Reading

For further reading on discrete probability distributions, we recommend the following resources:

  • Wikipedia: Discrete Probability Distribution
  • Khan Academy: Discrete Probability Distributions
  • MIT OpenCourseWare: Probability and Statistics

Glossary

  • Discrete probability distribution: A function that describes the probability of each possible outcome of a random variable.
  • Probability: A measure of the likelihood of an event occurring.
  • Random variable: A variable that takes on a value determined by chance.
  • Outcome: A possible value of a random variable.
    Discrete Probability Distribution: Q&A =====================================

Introduction

In our previous article, we discussed discrete probability distributions and completed a 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 outcome 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 include:

  • Non-negativity: The probability of each outcome must be non-negative.
  • Normalization: The sum of all probabilities must be equal to 1.
  • Countable outcomes: The number of possible outcomes must be countable.
  • Probability axioms: The probability distribution must satisfy the probability axioms, which include the axioms of non-negativity, normalization, and countable additivity.

Q: How do I determine the probability of an event in a discrete probability distribution?

A: To determine the probability of an event in a discrete probability distribution, you need to add up the probabilities of all the outcomes that satisfy the event.

Q: What is the difference between a discrete probability distribution and a continuous probability distribution?

A: A discrete probability distribution is a function that describes the probability of each possible outcome of a random variable, while a continuous probability distribution is a function that describes the probability of a range of values of a random variable.

Q: Can I use a discrete probability distribution to model a continuous random variable?

A: No, you cannot use a discrete probability distribution to model a continuous random variable. A discrete probability distribution is used to model a random variable that can take on a finite number of values, while a continuous probability distribution is used to model a random variable that can take on any value within a given range.

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

A: To find the expected value of a discrete random variable, you need to multiply each outcome by its probability and add up the results.

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 by taking the expected value of the squared difference between each outcome and the mean.

Q: Can I use a discrete probability distribution to model a real-world phenomenon?

A: Yes, you can use a discrete probability distribution to model a real-world phenomenon. Discrete probability distributions are used to model a wide range of real-world phenomena, including coin tossing, rolling a die, and random sampling.

Q: How do I choose the right discrete probability distribution for my data?

A: To choose the right discrete probability distribution for your data, you need to consider the characteristics of your data and the type of distribution that is most likely to model it.

Q: What are some common discrete probability distributions?

A: Some common discrete probability distributions include:

  • Bernoulli distribution: A distribution that models a binary random variable.
  • Binomial distribution: A distribution that models the number of successes in a fixed number of independent trials.
  • Poisson distribution: A distribution that models the number of events that occur in a fixed interval of time or space.

Conclusion

In this article, we answered some frequently asked questions about discrete probability distributions. We hope that this article has provided you with a better understanding of discrete probability distributions and how to use them to model real-world phenomena.

References

  • Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer.
  • Feller, W. (1950). An Introduction to Probability Theory and Its Applications. Wiley.
  • Ross, S. M. (2010). A First Course in Probability. Prentice Hall.

Further Reading

For further reading on discrete probability distributions, we recommend the following resources:

  • Wikipedia: Discrete Probability Distribution
  • Khan Academy: Discrete Probability Distributions
  • MIT OpenCourseWare: Probability and Statistics

Glossary

  • Discrete probability distribution: A function that describes the probability of each possible outcome of a random variable.
  • Probability: A measure of the likelihood of an event occurring.
  • Random variable: A variable that takes on a value determined by chance.
  • Outcome: A possible value of a random variable.