find the mean of the random variable x
Understand the Problem
The question is asking how to calculate the mean of a random variable x, which involves using the probability distribution associated with that random variable to find the average value.
Answer
Mean of a random variable $X$ is calculated using $E(X) = \sum (x_i \cdot P(X = x_i))$ for discrete or $E(X) = \int_{-\infty}^{+\infty} x \cdot f(x) \,dx$ for continuous.
Answer for screen readers
The mean of a random variable $X$ is calculated using the appropriate formula depending on whether it is discrete or continuous. For discrete variables, use:
$$ E(X) = \sum (x_i \cdot P(X = x_i)) $$
For continuous variables, use:
$$ E(X) = \int_{-\infty}^{+\infty} x \cdot f(x) ,dx $$
Steps to Solve
- Identify the Probability Distribution
Determine the probability distribution of the random variable $X$. This could be a discrete distribution (like a binomial or Poisson distribution) or a continuous distribution (like a normal distribution).
- List Possible Values and Probabilities
For a discrete random variable, list all possible values of $x_i$ and their corresponding probabilities $P(X = x_i)$. If $X$ is continuous, have the probability density function (pdf) available.
- Calculating the Mean for Discrete Random Variables
Use the formula for the mean:
$$ E(X) = \sum (x_i \cdot P(X = x_i)) $$
Calculate this sum by multiplying each possible value by its probability and then summing all these products.
- Calculating the Mean for Continuous Random Variables
If your random variable is continuous, use the following integral for mean:
$$ E(X) = \int_{-\infty}^{+\infty} x \cdot f(x) ,dx $$
where $f(x)$ is the probability density function of $X$. Evaluate this integral to find the mean.
- Final Calculation
Combine your results based on the type of distribution you have and compute the final result to determine the mean value of the random variable $X$.
The mean of a random variable $X$ is calculated using the appropriate formula depending on whether it is discrete or continuous. For discrete variables, use:
$$ E(X) = \sum (x_i \cdot P(X = x_i)) $$
For continuous variables, use:
$$ E(X) = \int_{-\infty}^{+\infty} x \cdot f(x) ,dx $$
More Information
The mean of a random variable gives us an idea of the average outcome of the random variable when observed over a large number of trials. In probability, the mean is often referred to as the expected value.
Tips
- Ignoring the proper formula for the type of random variable: Remember to use the summation for discrete random variables and integration for continuous ones.
- Forgetting to multiply each value by its probability when calculating the mean for discrete random variables.
- Miscalculating the probabilities such that they do not sum up to 1, which can lead to incorrect mean values.
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