Finding the Expected Value E(X) for Discrete Random Variables

Math Problem Statement

Consider the following data:

x −5 −4 −3 −2 −1 P(X=x) 0.2 0.3 0.2 0.1 0.2

Step 1 of 5 : Find the expected value E(X) . Round your answer to one decimal place.

The expected value $E(X)$ of a discrete random variable is calculated using the formula:

$E(X) = \sum x \cdot P(X = x)$

Where $x$ represents the values of the random variable, and $P(X = x)$ represents the probability associated with each value of $x$ .

Given the data:

\hline x & P(X=x) \\ \hline -5 & 0.2 \\ -4 & 0.3 \\ -3 & 0.2 \\ -2 & 0.1 \\ -1 & 0.2 \\ \hline \end{array}$$ We can now calculate the expected value $$E(X)$$: $$E(X) = (-5 \cdot 0.2) + (-4 \cdot 0.3) + (-3 \cdot 0.2) + (-2 \cdot 0.1) + (-1 \cdot 0.2)$$ Calculating each term: $$E(X) = (-1) + (-1.2) + (-0.6) + (-0.2) + (-0.2)$$ $$E(X) = -3.2$$ So, the expected value $$E(X)$$ is: $$E(X) = -3.2$$ This is the rounded value to one decimal place. Do you need further explanation, or should we proceed to the next step? Here are 5 related questions: 1. How do we interpret the expected value in terms of probability distributions? 2. What is the significance of the expected value for decision-making in probability theory? 3. How do we calculate the variance and standard deviation of a discrete random variable? 4. What is the difference between the expected value and the median in probability distributions? 5. How would the expected value change if the probabilities or the values of $$x$$ were different? **Tip:** The expected value gives the long-term average outcome of a random experiment when repeated many times.

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Discrete Probability Distribution
Expected Value
Probability Theory

E(X) = Σ x * P(X=x)

Expected Value Theorem

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