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.

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

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

Here, the values of \(Z\) and their corresponding probabilities are:

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

### Step-by-step calculation:

1. For \( Z = 1 \):
   \[
   (1 - 2)^2 = 1^2 = 1
   \]
   So, the contribution is:
   \[
   0.2 \times 1 = 0.2
   \]

2. For \( Z = 2 \):
   \[
   (2 - 2)^2 = 0^2 = 0
   \]
   So, the contribution is:
   \[
   0.4 \times 0 = 0
   \]

3. For \( Z = 3 \):
   \[
   (3 - 2)^2 = 1^2 = 1
   \]
   So, the contribution is:
   \[
   0.3 \times 1 = 0.3
   \]

4. For \( Z = 4 \):
   \[
   (4 - 2)^2 = 2^2 = 4
   \]
   So, the contribution is:
   \[
   0.1 \times 4 = 0.4
   \]

### Adding them all together:
\[
E([Z - 2]^2) = 0.2 + 0 + 0.3 + 0.4 = 0.9
\]

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

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

---

**Related questions:**

1. How would you compute the variance of \(Z\)?
2. What is the expected value \( E(Z) \)?
3. How can we interpret the result of \( E([Z - 2]^2) \) 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 \( E([Z - 1]^2) \)?

**Tip:** For calculating expected values of functions of random variables, always remember to expand the function first (e.g., \( (Z - 2)^2 \)) and then apply the formula for the expected value.

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.

Learn how to compute the expected value E([Z−2]^2) for a discrete random variable Z with a given probability distribution. This problem explores key probability concepts such as expected value and variance, suitable for high school students studying probability.

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