Given The Discrete Random Variable X X X , What Is The Probability Distribution Of X X X If X \sim B\left(2, \frac{5}{17}\right ]?A. ${ \begin{array}{l} P(X=0)=0.498 \ P(X=1)=0.415 \ P(X=2)=0.087 \end{array} }$B.

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 explore the probability distribution of a discrete random variable $X$ that follows a binomial distribution with parameters $n=2$ and $p=\frac{5}{17}$.

The Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. The probability mass function of the binomial distribution is given by:

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where $n$ is the number of trials, $p$ is the probability of success, and $k$ is the number of successes.

The Given Problem

In this problem, we are given that $X \sim B\left(2, \frac{5}{17}\right)$. This means that $X$ follows a binomial distribution with $n=2$ and $p=\frac{5}{17}$. We need to find the probability distribution of $X$, which is the function that assigns a probability to each possible value of $X$.

Calculating the Probability Distribution

To calculate the probability distribution of $X$, we need to use the formula for the binomial distribution:

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

In this case, $n=2$ and $p=\frac{5}{17}$. We need to calculate the probability of $X$ taking on the values $0$, $1$, and $2$.

Calculating the Probability of $X=0$

To calculate the probability of $X=0$, we need to use the formula:

$$P(X=0) = \binom{n}{0} p^0 (1-p)^{n-0}$$

Substituting $n=2$ and $p=\frac{5}{17}$, we get:

$$P(X=0) = \binom{2}{0} \left(\frac{5}{17}\right)^0 \left(1-\frac{5}{17}\right)^{2-0}$$

Simplifying, we get:

$$P(X=0) = 1 \times 1 \times \left(\frac{12}{17}\right)^2$$

$$P(X=0) = \frac{144}{289}$$

Calculating the Probability of $X=1$

To calculate the probability of $X=1$, we need to use the formula:

$$P(X=1) = \binom{n}{1} p^1 (1-p)^{n-1}$$

Substituting $n=2$ and $p=\frac{5}{17}$, we get:

$$P(X=1) = \binom{2}{1} \left(\frac{5}{17}\right)^1 \left(1-\frac{5}{17}\right)^{2-1}$$

Simplifying, we get:

$$P(X=1) = 2 \times \frac{5}{17} \times \frac{12}{17}$$

$$P(X=1) = \frac{120}{289}$$

Calculating the Probability of $X=2$

To calculate the probability of $X=2$, we need to use the formula:

$$P(X=2) = \binom{n}{2} p^2 (1-p)^{n-2}$$

Substituting $n=2$ and $p=\frac{5}{17}$, we get:

$$P(X=2) = \binom{2}{2} \left(\frac{5}{17}\right)^2 \left(1-\frac{5}{17}\right)^{2-2}$$

Simplifying, we get:

$$P(X=2) = 1 \times \left(\frac{5}{17}\right)^2 \times 1$$

$$P(X=2) = \frac{25}{289}$$

Conclusion

In this article, we have calculated the probability distribution of a discrete random variable $X$ that follows a binomial distribution with parameters $n=2$ and $p=\frac{5}{17}$. We have found that the probability distribution of $X$ is given by:

$$P(X=0) = 0.498$$

$$P(X=1) = 0.415$$

$$P(X=2) = 0.087$$

This result shows that the probability of $X$ taking on the value $0$ is the highest, followed by the probability of $X$ taking on the value $1$, and then the probability of $X$ taking on the value $2$.

Discussion

The binomial distribution is a widely used probability distribution in statistics and engineering. It is used to model the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. The binomial distribution is a discrete probability distribution, which means that it can only take on a countable number of distinct values.

In this problem, we have used the binomial distribution to model the probability distribution of a discrete random variable $X$. We have found that the probability distribution of $X$ is given by:

$$P(X=0) = 0.498$$

$$P(X=1) = 0.415$$

$$P(X=2) = 0.087$$

This result shows that the probability of $X$ taking on the value $0$ is the highest, followed by the probability of $X$ taking on the value $1$, and then the probability of $X$ taking on the value $2$.

References

• [1] Johnson, N. L., & Kotz, S. (1969). Distributions in statistics: discrete distributions. Wiley.
• [2] Feller, W. (1968). An introduction to probability theory and its applications. Wiley.

Appendix

The following is a summary of the calculations performed in this article:

Value of $X$	Probability of $X$
0	0.498
1	0.415
2	0.087

This summary shows the probability distribution of $X$, which is the function that assigns a probability to each possible value of $X$.

Introduction

In our previous article, we explored the probability distribution of a discrete random variable $X$ that follows a binomial distribution with parameters $n=2$ and $p=\frac{5}{17}$. We calculated the probability distribution of $X$ and found that the probability of $X$ taking on the value $0$ is the highest, followed by the probability of $X$ taking on the value $1$, and then the probability of $X$ taking on the value $2$.

In this article, we will answer some frequently asked questions (FAQs) about the probability distribution of a discrete random variable.

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. In other words, it is a way of describing the probability of each possible outcome of a random experiment.

Q: What is the binomial distribution?

A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

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

A: To calculate the probability distribution of a discrete random variable, you need to use the formula for the binomial distribution:

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where $n$ is the number of trials, $p$ is the probability of success, and $k$ is the number of successes.

Q: What is the probability of $X=0$?

A: The probability of $X=0$ is given by:

$$P(X=0) = \binom{n}{0} p^0 (1-p)^{n-0}$$

Substituting $n=2$ and $p=\frac{5}{17}$, we get:

$$P(X=0) = \frac{144}{289}$$

Q: What is the probability of $X=1$?

A: The probability of $X=1$ is given by:

$$P(X=1) = \binom{n}{1} p^1 (1-p)^{n-1}$$

Substituting $n=2$ and $p=\frac{5}{17}$, we get:

$$P(X=1) = \frac{120}{289}$$

Q: What is the probability of $X=2$?

A: The probability of $X=2$ is given by:

$$P(X=2) = \binom{n}{2} p^2 (1-p)^{n-2}$$

Substituting $n=2$ and $p=\frac{5}{17}$, we get:

$$P(X=2) = \frac{25}{289}$$

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

A: The probability distribution of a discrete random variable can be used in a variety of real-world applications, such as:

• Modeling the number of successes in a fixed number of independent trials
• Predicting the probability of a particular outcome in a random experiment
• Making decisions based on the probability of different outcomes

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about the probability distribution of a discrete random variable. We have also provided a summary of the calculations performed in our previous article, which showed the probability distribution of a discrete random variable $X$ that follows a binomial distribution with parameters $n=2$ and $p=\frac{5}{17}$.

We hope that this article has been helpful in answering your questions about the probability distribution of a discrete random variable. If you have any further questions, please don't hesitate to contact us.

References

• [1] Johnson, N. L., & Kotz, S. (1969). Distributions in statistics: discrete distributions. Wiley.
• [2] Feller, W. (1968). An introduction to probability theory and its applications. Wiley.

Appendix

The following is a summary of the calculations performed in this article:

Value of $X$	Probability of $X$
0	0.498
1	0.415
2	0.087

This summary shows the probability distribution of $X$, which is the function that assigns a probability to each possible value of $X$.