\begin{tabular}{c|ccccc}$X$ & 0 & 1 & 2 & 3 & 4 \\hline $P ( X )$ & 0.4 & 0.2 & 0.1 & 0.2 & 0.1\end{tabular}Find The Mean. Round Your Answer To One Decimal Place As Needed.Mean: $\mu =
Understanding the Problem
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 discuss how to calculate the mean of a discrete random variable.
What is the Mean?
The mean, also known as the expected value, is a measure of the central tendency of a probability distribution. It represents the long-run average value that the random variable is expected to take on. The mean is calculated by multiplying each possible value of the random variable by its probability and summing the results.
Calculating the Mean
To calculate the mean of a discrete random variable, we need to follow these steps:
- List all the possible values of the random variable and their corresponding probabilities.
- Multiply each possible value by its probability.
- Sum the results of the multiplications.
Example
Let's consider the probability distribution of a discrete random variable X, which is given in the table below:
| X | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P(X) | 0.4 | 0.2 | 0.1 | 0.2 | 0.1 |
To calculate the mean, we need to multiply each possible value of X by its probability and sum the results.
| X | P(X) | X * P(X) |
|---|---|---|
| 0 | 0.4 | 0 |
| 1 | 0.2 | 0.2 |
| 2 | 0.1 | 0.2 |
| 3 | 0.2 | 0.6 |
| 4 | 0.1 | 0.4 |
Now, we sum the results of the multiplications:
0 + 0.2 + 0.2 + 0.6 + 0.4 = 1.4
Therefore, the mean of the discrete random variable X is 1.4.
Rounding the Answer
Since the problem asks us to round the answer to one decimal place, we need to round 1.4 to one decimal place. The rounded answer is 1.4.
Conclusion
In this article, we discussed how to calculate the mean of a discrete random variable. We listed the steps to calculate the mean, and we used an example to illustrate the process. We also discussed the importance of rounding the answer to one decimal place, as required by the problem.
Mean Formula
The mean of a discrete random variable X is calculated using the following formula:
where x is the possible value of the random variable, P(x) is the probability of the value x, and the sum is taken over all possible values of the random variable.
Properties of the Mean
The mean has several important properties, including:
- Linearity: The mean is a linear function of the random variable. This means that if we multiply the random variable by a constant, the mean is also multiplied by the same constant.
- Homogeneity: The mean is homogeneous of degree 1. This means that if we multiply the random variable by a constant, the mean is also multiplied by the same constant.
- Additivity: The mean is additive. This means that if we have two independent random variables, the mean of the sum of the two variables is equal to the sum of the means of the two variables.
Real-World Applications
The mean is an important concept in many real-world applications, including:
- Finance: The mean is used to calculate the expected return on investment of a portfolio of stocks or bonds.
- Insurance: The mean is used to calculate the expected payout of an insurance policy.
- Engineering: The mean is used to calculate the expected value of a system or a process.
Conclusion
Q: What is the mean of a discrete random variable?
A: The mean of a discrete random variable is a measure of the central tendency of the probability distribution. It represents the long-run average value that the random variable is expected to take on.
Q: How is the mean calculated?
A: The mean is calculated by multiplying each possible value of the random variable by its probability and summing the results.
Q: What is the formula for calculating the mean?
A: The formula for calculating the mean is:
where x is the possible value of the random variable, P(x) is the probability of the value x, and the sum is taken over all possible values of the random variable.
Q: What are the properties of the mean?
A: The mean has several important properties, including:
- Linearity: The mean is a linear function of the random variable. This means that if we multiply the random variable by a constant, the mean is also multiplied by the same constant.
- Homogeneity: The mean is homogeneous of degree 1. This means that if we multiply the random variable by a constant, the mean is also multiplied by the same constant.
- Additivity: The mean is additive. This means that if we have two independent random variables, the mean of the sum of the two variables is equal to the sum of the means of the two variables.
Q: What are some real-world applications of the mean?
A: The mean is used in many real-world applications, including:
- Finance: The mean is used to calculate the expected return on investment of a portfolio of stocks or bonds.
- Insurance: The mean is used to calculate the expected payout of an insurance policy.
- Engineering: The mean is used to calculate the expected value of a system or a process.
Q: How do I round the mean to one decimal place?
A: To round the mean to one decimal place, you need to look at the second decimal place. If the second decimal place is 5 or greater, you round up. If the second decimal place is less than 5, you round down.
Q: What is the difference between the mean and the median?
A: The mean and the median are both measures of central tendency, but they are calculated differently. The mean is calculated by multiplying each possible value of the random variable by its probability and summing the results. The median is the middle value of the data when it is arranged in order.
Q: Can the mean be negative?
A: Yes, the mean can be negative. This occurs when the random variable takes on negative values and the probability of these values is greater than 0.
Q: Can the mean be zero?
A: Yes, the mean can be zero. This occurs when the random variable takes on zero and the probability of this value is greater than 0.
Q: Can the mean be greater than 1?
A: Yes, the mean can be greater than 1. This occurs when the random variable takes on values greater than 1 and the probability of these values is greater than 0.
Q: How do I calculate the mean of a continuous random variable?
A: The mean of a continuous random variable is calculated using the following formula:
where x is the possible value of the random variable, f(x) is the probability density function of the random variable, and the integral is taken over all possible values of the random variable.
Q: What is the difference between the mean of a discrete random variable and the mean of a continuous random variable?
A: The mean of a discrete random variable is calculated by multiplying each possible value of the random variable by its probability and summing the results. The mean of a continuous random variable is calculated using the integral of the product of the random variable and its probability density function.