Math Problem Statement
The discrete random variable Z has the probability distribution
1 2 3 4
0.2 0.4 0.3 0.1
What is E([Z−2]^2) ?
Give your answer as an exact whole number or decimal.
Solution
To calculate , we follow the formula for the expected value of a function of a discrete random variable. Specifically:
Here, the values of and their corresponding probabilities are:
- ,
- ,
- ,
- ,
Step-by-step calculation:
-
For : So, the contribution is:
-
For : So, the contribution is:
-
For : So, the contribution is:
-
For : So, the contribution is:
Adding them all together:
So, the expected value is 0.9.
Let me know if you need further clarification or additional details!
Related questions:
- How would you compute the variance of ?
- What is the expected value ?
- How can we interpret the result of in terms of variance?
- What is the probability distribution of a random variable, and how do you use it in these calculations?
- Can we compute other similar expressions, such as ?
Tip: For calculating expected values of functions of random variables, always remember to expand the function first (e.g., ) and then apply the formula for the expected value.
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Math Problem Analysis
Mathematical Concepts
Probability
Expected Value
Variance of Discrete Random Variables
Formulas
E[g(Z)] = Σ P(Z = z) * g(z)
Variance formula: Var(Z) = E[(Z - μ)^2]
Theorems
Expected Value Theorem
Variance Definition
Suitable Grade Level
Grades 10-12
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