Math Problem Statement

Consider the following data:

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

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

Solution

To find the expected value E(X)E(X) of a discrete random variable XX, we use the formula:

E(X)=∑xx⋅P(X=x)E(X) = \sum_{x} x \cdot P(X = x)

Where:

  • xx are the values of the random variable,
  • P(X=x)P(X = x) are the corresponding probabilities.

Given data:

\hline x & P(X = x) \\ \hline 1 & 0.3 \\ 2 & 0.1 \\ 3 & 0.2 \\ 4 & 0.1 \\ 5 & 0.3 \\ \hline \end{array}$$ Now, applying the formula: $$E(X) = (1 \times 0.3) + (2 \times 0.1) + (3 \times 0.2) + (4 \times 0.1) + (5 \times 0.3)$$ $$E(X) = 0.3 + 0.2 + 0.6 + 0.4 + 1.5$$ $$E(X) = 3.0$$ So, the expected value $$E(X) = 3.0$$. Would you like me to continue with further details or answer any questions? Here are some related questions you might find interesting: 1. How do you calculate the variance of a discrete random variable? 2. What is the difference between expected value and variance? 3. How would you find the standard deviation of the given data? 4. What happens if the probabilities do not sum to 1 in a probability distribution? 5. How would the expected value change if the probability distribution was different? Tip: To help ensure your calculations are correct, always double-check that the sum of the probabilities equals 1.

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Math Problem Analysis

Mathematical Concepts

Expected Value
Discrete Random Variable
Probability Distribution

Formulas

E(X) = Σ(x * P(X = x))

Theorems

Law of Total Probability

Suitable Grade Level

Grades 9-12

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