Consider The Discrete Random Variable { X $}$ Given In The Table Below. Calculate The Mean, Variance, And Standard Deviation Of { X $}$. Also, Calculate The Expected Value Of { X $}$. Round Your Solution To Three
Introduction
In probability theory, a discrete random variable is a variable that can take on a countable number of distinct values. In this article, we will consider a discrete random variable { X $}$ given in the table below and calculate its mean, variance, and standard deviation. We will also calculate the expected value of { X $}$.
Table of Values
| Value of X | Probability |
|---|---|
| 1 | 0.2 |
| 2 | 0.3 |
| 3 | 0.4 |
| 4 | 0.1 |
Calculating the Expected Value
The expected value of a discrete random variable is calculated by multiplying each value of the variable by its probability and summing the results.
Expected Value Formula
E(X) = ∑xP(x)
where E(X) is the expected value, x is the value of the variable, and P(x) is the probability of the value.
Calculating the Expected Value of { X $}$
Using the table of values, we can calculate the expected value of { X $}$ as follows:
E(X) = (1)(0.2) + (2)(0.3) + (3)(0.4) + (4)(0.1) = 0.2 + 0.6 + 1.2 + 0.4 = 2.4
Calculating the Mean
The mean of a discrete random variable is equal to its expected value.
Mean Formula
μ = E(X)
where μ is the mean and E(X) is the expected value.
Calculating the Mean of { X $}$
Using the expected value calculated above, we can calculate the mean of { X $}$ as follows:
μ = E(X) = 2.4
Calculating the Variance
The variance of a discrete random variable is calculated by finding the average of the squared differences between each value of the variable and the mean.
Variance Formula
σ^2 = ∑(x-μ)^2P(x)
where σ^2 is the variance, x is the value of the variable, μ is the mean, and P(x) is the probability of the value.
Calculating the Variance of { X $}$
Using the table of values and the mean calculated above, we can calculate the variance of { X $}$ as follows:
σ^2 = (1-2.4)^2 + (2-2.4)^2 + (3-2.4)^2 + (4-2.4)^2 = (1.4)^2 + (0.4)^2 + (0.6)^2 + (1.6)^2 = 1.96 + 0.048 + 0.144 + 0.256 = 2.408
Calculating the Standard Deviation
The standard deviation of a discrete random variable is the square root of its variance.
Standard Deviation Formula
σ = √σ^2
where σ is the standard deviation and σ^2 is the variance.
Calculating the Standard Deviation of { X $}$
Using the variance calculated above, we can calculate the standard deviation of { X $}$ as follows:
σ = √2.408 = 1.553
Conclusion
In this article, we calculated the mean, variance, and standard deviation of a discrete random variable { X $}$ given in the table below. We also calculated the expected value of { X $}$. The results are as follows:
- Expected Value: 2.4
- Mean: 2.4
- Variance: 2.408
- Standard Deviation: 1.553
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 only take on a specific set of values, and each value has a specific probability of occurring.
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 value of the variable by its probability and sum the results. The formula for the expected value is:
E(X) = ∑xP(x)
where E(X) is the expected value, x is the value of the variable, and P(x) is the probability of the value.
Q: What is the difference between the mean and the expected value?
A: The mean and the expected value are actually the same thing. The mean is the average value of the variable, and the expected value is the long-term average value of the variable. They are calculated using the same formula:
μ = E(X)
where μ is the mean and E(X) is the expected value.
Q: How do I calculate the variance of a discrete random variable?
A: To calculate the variance of a discrete random variable, you need to find the average of the squared differences between each value of the variable and the mean. The formula for the variance is:
σ^2 = ∑(x-μ)^2P(x)
where σ^2 is the variance, x is the value of the variable, μ is the mean, and P(x) is the probability of the value.
Q: What is the standard deviation, and how do I calculate it?
A: The standard deviation is the square root of the variance. It is a measure of the spread or dispersion of the variable. To calculate the standard deviation, you need to take the square root of the variance:
σ = √σ^2
where σ is the standard deviation and σ^2 is the variance.
Q: Why is it important to calculate the mean, variance, and standard deviation of a discrete random variable?
A: Calculating the mean, variance, and standard deviation of a discrete random variable is important because it helps you understand the distribution of the variable and make predictions about its behavior. It also helps you to identify the spread or dispersion of the variable, which is useful in many applications.
Q: Can I use the same formulas to calculate the mean, variance, and standard deviation of a continuous random variable?
A: No, the formulas for calculating the mean, variance, and standard deviation of a continuous random variable are different from those for a discrete random variable. For a continuous random variable, you need to use the following formulas:
μ = ∫xf(x)dx
σ^2 = ∫(x-μ)^2f(x)dx
σ = √σ^2
where μ is the mean, σ^2 is the variance, σ is the standard deviation, x is the value of the variable, f(x) is the probability density function, and dx is the infinitesimal change in x.
Q: Where can I find more information about discrete random variables and probability theory?
A: There are many resources available online and in textbooks that can help you learn more about discrete random variables and probability theory. Some popular resources include:
- Khan Academy: Probability and Statistics
- MIT OpenCourseWare: Probability and Statistics
- Wikipedia: Probability Theory
- Wolfram MathWorld: Probability Theory
I hope this FAQ article has been helpful in answering your questions about discrete random variables and probability theory. If you have any more questions, feel free to ask!