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This problem presents the probability density function (PDF) of a random variable \( X \) as:

\[
f(x) = A e^{-x^2 + 2x - 1}
\]

We are tasked with determining the correct conclusions about the expected value \( E(X) \) and the variance \( D(X) \) of the random variable \( X \). The options for \( E(X) \) and \( D(X) \) are provided as:

- (A) \( E(X) = 1, D(X) = \frac{1}{2} \)
- (B) \( E(X) = 1, D(X) = 2 \)
- (C) \( E(X) = -1, D(X) = 2 \)
- (D) \( E(X) = -1, D(X) = \frac{1}{2} \)

### Steps to solve:

1. **Rewrite the exponent in standard form**:
   The exponent of the given PDF can be simplified:
   \[
   -x^2 + 2x - 1 = -(x^2 - 2x + 1) = -(x - 1)^2
   \]
   So the PDF becomes:
   \[
   f(x) = A e^{-(x-1)^2}
   \]
   This is the PDF of a normal distribution, where the general form is:
   \[
   f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
   \]
   By comparison, the given PDF corresponds to a normal distribution with:
   - Mean \( \mu = 1 \)
   - Variance \( \sigma^2 = 1/2 \)

2. **Determine the correct values**:
   From the comparison:
   - The expected value \( E(X) = 1 \)
   - The variance \( D(X) = \frac{1}{2} \)

Thus, the correct answer is **(A)**: \( E(X) = 1 \), \( D(X) = \frac{1}{2} \).

Would you like a more detailed explanation or have any questions? Here are five related questions to expand on this topic:

1. How is the expected value of a normal distribution calculated?
2. What are the properties of the variance in a probability distribution?
3. Can you explain the role of the constant \( A \) in this probability density function?
4. How would you normalize the given probability density function \( f(x) \)?
5. How does the standard deviation relate to variance in a normal distribution?

**Tip:** The exponent of a probability density function for a normal distribution is always expressed as a quadratic term involving \( x \), shifted by the mean.

已知随机向量X的密度函数为 f(x)=A e^(-x^2 + 2x - 1)，则下列结论正确的是（A）EX=1,DX=1/2；（B）EX=1,DX=2；（C）EX=-1,DX=2；（D）EX=-1,DX=1/2

Solve a probability problem to determine the expected value (E(X)) and variance (D(X)) of a random variable X, using the PDF f(x)=Ae^(-x^2 + 2x - 1). The solution applies concepts from normal distributions and involves manipulating quadratic expressions. Suitable for undergraduate students in statistics.

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