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 E([Z2]2)E([Z - 2]^2), we follow the formula for the expected value of a function of a discrete random variable. Specifically:

E([Z2]2)=zP(Z=z)(z2)2E([Z - 2]^2) = \sum_{z} P(Z = z) \cdot (z - 2)^2

Here, the values of ZZ and their corresponding probabilities are:

  • Z=1Z = 1, P(Z=1)=0.2P(Z = 1) = 0.2
  • Z=2Z = 2, P(Z=2)=0.4P(Z = 2) = 0.4
  • Z=3Z = 3, P(Z=3)=0.3P(Z = 3) = 0.3
  • Z=4Z = 4, P(Z=4)=0.1P(Z = 4) = 0.1

Step-by-step calculation:

  1. For Z=1Z = 1: (12)2=12=1(1 - 2)^2 = 1^2 = 1 So, the contribution is: 0.2×1=0.20.2 \times 1 = 0.2

  2. For Z=2Z = 2: (22)2=02=0(2 - 2)^2 = 0^2 = 0 So, the contribution is: 0.4×0=00.4 \times 0 = 0

  3. For Z=3Z = 3: (32)2=12=1(3 - 2)^2 = 1^2 = 1 So, the contribution is: 0.3×1=0.30.3 \times 1 = 0.3

  4. For Z=4Z = 4: (42)2=22=4(4 - 2)^2 = 2^2 = 4 So, the contribution is: 0.1×4=0.40.1 \times 4 = 0.4

Adding them all together:

E([Z2]2)=0.2+0+0.3+0.4=0.9E([Z - 2]^2) = 0.2 + 0 + 0.3 + 0.4 = 0.9

So, the expected value E([Z2]2)E([Z - 2]^2) is 0.9.

Let me know if you need further clarification or additional details!


Related questions:

  1. How would you compute the variance of ZZ?
  2. What is the expected value E(Z)E(Z)?
  3. How can we interpret the result of E([Z2]2)E([Z - 2]^2) in terms of variance?
  4. What is the probability distribution of a random variable, and how do you use it in these calculations?
  5. Can we compute other similar expressions, such as E([Z1]2)E([Z - 1]^2)?

Tip: For calculating expected values of functions of random variables, always remember to expand the function first (e.g., (Z2)2(Z - 2)^2) 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