Math Problem Statement

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.

Solution

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

1. E[X]=1λ\mathbb{E}[X] = \frac{1}{\lambda}

  • True: The expected value of an exponential random variable XX with rate parameter λ\lambda is given by E[X]=1λ\mathbb{E}[X] = \frac{1}{\lambda}.

2. Var(X)=1λ2\text{Var}(X) = \frac{1}{\lambda^2}

  • True: The variance of an exponential random variable XX with rate parameter λ\lambda is given by Var(X)=1λ2\text{Var}(X) = \frac{1}{\lambda^2}.

3. P(X>x+kX>k)=P(X>x)P(X > x + k \mid X > k) = P(X > x) for k,x0k, 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+kx + k given that it has already exceeded kk is the same as the probability that it exceeds xx from the start.

4. P(X>x+kX>k)=P(X>k)P(X > x + k \mid X > k) = P(X > k) for k,x0k, x \geq 0

  • False: This statement is incorrect. According to the memoryless property, P(X>x+kX>k)=P(X>x)P(X > x + k \mid X > k) = P(X > x), not P(X>k)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 λ\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.

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Math Problem Analysis

Mathematical Concepts

Exponential random variable
Probability

Formulas

Expected value of an exponential random variable
Variance of an exponential random variable
Memoryless property of exponential distribution

Theorems

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Suitable Grade Level

Undergraduate