1 point
If
X
X is an exponential random variable with rate parameter
λ
λ then which of the following statement(s) is(are) correct.

E
[
X
]
E[X] =
1
λ
λ
1
​

V
a
r
[
X
]
Var[X] =
1
λ
2
λ
2

1
​

P
(
X
>
x
+
k
∣
X
>
k
)
P(X>x+k∣X>k) =
P
(
X
>
x
)
P(X>x) for
k
,
x
≥
0
k,x≥0.

P
(
X
>
x
+
k
∣
X
>
k
)
P(X>x+k∣X>k) =
P
(
X
>
k
)
P(X>k) for
k
,
x
≥
0
k,x≥0.

Given that \( X \) is an exponential random variable with rate parameter \( \lambda \), let's analyze each of the statements to determine which are correct.

### 1. \( \mathbb{E}[X] = \frac{1}{\lambda} \)
- **True**: The expected value of an exponential random variable \( X \) with rate parameter \( \lambda \) is given by \( \mathbb{E}[X] = \frac{1}{\lambda} \).

### 2. \( \text{Var}(X) = \frac{1}{\lambda^2} \)
- **True**: The variance of an exponential random variable \( X \) with rate parameter \( \lambda \) is given by \( \text{Var}(X) = \frac{1}{\lambda^2} \).

### 3. \( P(X > x + k \mid X > k) = P(X > x) \) for \( k, x \geq 0 \)
- **True**: This property is known as the **memoryless property** of the exponential distribution. It states that the probability that the process exceeds \( x + k \) given that it has already exceeded \( k \) is the same as the probability that it exceeds \( x \) from the start.

### 4. \( P(X > x + k \mid X > k) = P(X > k) \) for \( k, x \geq 0 \)
- **False**: This statement is incorrect. According to the memoryless property, \( P(X > x + k \mid X > k) = P(X > x) \), not \( P(X > k) \).

### Conclusion:
- Statements 1, 2, and 3 are correct.
- Statement 4 is incorrect.

Would you like more details or have any questions?

Here are some related questions:

1. What is the probability density function (PDF) of an exponential distribution?
2. How does the memoryless property apply to other distributions, if at all?
3. Can you derive the variance of an exponential distribution?
4. What are some real-world scenarios where the exponential distribution is used?
5. How does the rate parameter \( \lambda \) affect the shape of the exponential distribution?

**Tip:** The memoryless property is unique to the exponential distribution among continuous distributions, making it useful in modeling scenarios with no aging effect, such as the lifetime of electronic components.

Explore the properties of exponential random variables with examples. Learn about the expected value, variance, and the memoryless property of the exponential distribution. Suitable for undergraduate students and those interested in probability theory.

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