Use The PDF For The Random Variable, \[$ X = \$\] Score On Quiz 8, To Answer The Questions.$\[ \begin{array}{|c|c|} \hline x & P(X=x) \\ \hline 0 & 0.001 \\ \hline 1 & 0.002 \\ \hline 2 & 0.004 \\ \hline 3 & 0.095 \\ \hline 4 & 0.105

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Introduction

In probability theory, a random variable is a variable whose possible values are determined by chance events. The probability distribution of a random variable is a function that describes the probability of each possible value of the variable. In this article, we will use the probability distribution of a random variable, denoted as X, to answer questions related to its properties and behavior.

The Probability Distribution of X

The probability distribution of X is given in the following table:

x P(X=x)
0 0.001
1 0.002
2 0.004
3 0.095
4 0.105

Calculating the Mean of X

The mean of a random variable is a measure of its central tendency. It is calculated by multiplying each possible value of the variable by its probability and summing the results. In this case, the mean of X can be calculated as follows:

μ=∑x=04xP(X=x)\mu = \sum_{x=0}^{4} xP(X=x)

μ=0(0.001)+1(0.002)+2(0.004)+3(0.095)+4(0.105)\mu = 0(0.001) + 1(0.002) + 2(0.004) + 3(0.095) + 4(0.105)

μ=0+0.002+0.008+0.285+0.42\mu = 0 + 0.002 + 0.008 + 0.285 + 0.42

μ=0.715\mu = 0.715

Calculating the Variance of X

The variance of a random variable is a measure of its spread or dispersion. It is calculated by finding the average of the squared differences between each possible value of the variable and its mean. In this case, the variance of X can be calculated as follows:

σ2=∑x=04(x−μ)2P(X=x)\sigma^2 = \sum_{x=0}^{4} (x-\mu)^2P(X=x)

σ2=(0−0.715)2(0.001)+(1−0.715)2(0.002)+(2−0.715)2(0.004)+(3−0.715)2(0.095)+(4−0.715)2(0.105)\sigma^2 = (0-0.715)^2(0.001) + (1-0.715)^2(0.002) + (2-0.715)^2(0.004) + (3-0.715)^2(0.095) + (4-0.715)^2(0.105)

σ2=0.515+0.102+0.204+1.351+1.351\sigma^2 = 0.515 + 0.102 + 0.204 + 1.351 + 1.351

σ2=3.523\sigma^2 = 3.523

Calculating the Standard Deviation of X

The standard deviation of a random variable is the square root of its variance. In this case, the standard deviation of X can be calculated as follows:

σ=σ2\sigma = \sqrt{\sigma^2}

σ=3.523\sigma = \sqrt{3.523}

σ=1.877\sigma = 1.877

Calculating the Probability of X Being Greater Than 2

The probability of X being greater than 2 can be calculated by summing the probabilities of X being equal to 3 and 4.

P(X>2)=P(X=3)+P(X=4)P(X>2) = P(X=3) + P(X=4)

P(X>2)=0.095+0.105P(X>2) = 0.095 + 0.105

P(X>2)=0.2P(X>2) = 0.2

Calculating the Probability of X Being Less Than 2

The probability of X being less than 2 can be calculated by summing the probabilities of X being equal to 0 and 1.

P(X<2)=P(X=0)+P(X=1)P(X<2) = P(X=0) + P(X=1)

P(X<2)=0.001+0.002P(X<2) = 0.001 + 0.002

P(X<2)=0.003P(X<2) = 0.003

Conclusion

In this article, we used the probability distribution of a random variable, denoted as X, to answer questions related to its properties and behavior. We calculated the mean, variance, and standard deviation of X, as well as the probability of X being greater than 2 and less than 2. These calculations demonstrate the importance of understanding the probability distribution of a random variable in probability theory.

References

  • [1] Ross, S. M. (2014). Introduction to probability models. Academic Press.
  • [2] Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions. John Wiley & Sons.

Appendix

The following table summarizes the calculations performed in this article:

Calculation Result
Mean of X 0.715
Variance of X 3.523
Standard Deviation of X 1.877
Probability of X being greater than 2 0.2
Probability of X being less than 2 0.003

Q: What is the probability distribution of X?

A: The probability distribution of X is a function that describes the probability of each possible value of the variable X. It is given in the following table:

x P(X=x)
0 0.001
1 0.002
2 0.004
3 0.095
4 0.105

Q: What is the mean of X?

A: The mean of X is a measure of its central tendency. It is calculated by multiplying each possible value of the variable by its probability and summing the results. In this case, the mean of X is 0.715.

Q: What is the variance of X?

A: The variance of X is a measure of its spread or dispersion. It is calculated by finding the average of the squared differences between each possible value of the variable and its mean. In this case, the variance of X is 3.523.

Q: What is the standard deviation of X?

A: The standard deviation of X is the square root of its variance. In this case, the standard deviation of X is 1.877.

Q: What is the probability of X being greater than 2?

A: The probability of X being greater than 2 can be calculated by summing the probabilities of X being equal to 3 and 4. In this case, the probability of X being greater than 2 is 0.2.

Q: What is the probability of X being less than 2?

A: The probability of X being less than 2 can be calculated by summing the probabilities of X being equal to 0 and 1. In this case, the probability of X being less than 2 is 0.003.

Q: How do I calculate the probability of X being within a certain range?

A: To calculate the probability of X being within a certain range, you can sum the probabilities of X being equal to each value within the range. For example, to calculate the probability of X being between 1 and 3, you would sum the probabilities of X being equal to 1, 2, and 3.

Q: How do I calculate the probability of X being outside a certain range?

A: To calculate the probability of X being outside a certain range, you can subtract the probability of X being within the range from 1. For example, to calculate the probability of X being outside the range of 1 to 3, you would subtract the probability of X being between 1 and 3 from 1.

Q: What is the relationship between the mean and variance of X?

A: The mean and variance of X are related in that the variance is the average of the squared differences between each possible value of the variable and its mean. This means that the variance is always greater than or equal to 0, and the mean is always less than or equal to the variance.

Q: How do I use the probability distribution of X to make predictions?

A: To use the probability distribution of X to make predictions, you can use the probabilities of X being equal to each value to estimate the likelihood of different outcomes. For example, if you want to predict the probability of X being greater than 2, you can use the probability of X being equal to 3 and 4 to estimate the likelihood of this outcome.

Q: What are some common applications of the probability distribution of X?

A: The probability distribution of X has many common applications in fields such as finance, engineering, and economics. For example, it can be used to model the behavior of stock prices, predict the likelihood of natural disasters, and estimate the probability of different outcomes in a game or competition.

Q: How do I interpret the results of a probability distribution?

A: To interpret the results of a probability distribution, you need to understand the meaning of the probabilities and how they relate to the variable X. You should also consider the context in which the probability distribution is being used, as this can affect the interpretation of the results.