Question

Let X be an exp. distributed random variable with mean $\lambda$($>$ 0) if P (X$>$ 5) = 0.35 then the conditional probability P(x$>$ 10$|$ x$>$ 5) is _______.
 

Hint:

Recognizing the memoryless property is the key to solving this type of problem instantly. If you see a conditional probability question of the form P(X> s+t | X> s) for an exponential distribution, the answer is always P(X> t).

Answer 1

Correct Answer: 0.35

Step 1: Understanding the Question:
We are given an exponentially distributed random variable X. We are given the probability P(X>5) and asked to find the conditional probability P(X>10 | X>5). This question is a direct test of the memoryless property of the exponential distribution.
Step 2: Key Formula or Approach:
The exponential distribution has a unique property called the memoryless property. It states that for any s, t> 0: \[ P(X>s + t \mid X>s) = P(X>t) \] In simple terms, the probability that the event will happen in the future is independent of how much time has already passed.
Step 3: Detailed Explanation:
We need to calculate P(X> 10 | X> 5).
We can rewrite this in the form of the memoryless property, with s = 5 and t = 5: \[ P(X>5 + 5 \mid X>5) \] According to the memoryless property, this is equal to: \[ P(X>5) \] The problem statement gives us the value of P(X> 5). \[ P(X>5) = 0.35 \] Therefore, \[ P(X>10 \mid X>5) = P(X>5) = 0.35 \] Alternative Calculation (without direct use of the property):
The cumulative distribution function (CDF) for an exponential distribution is $F(x) = P(X \le x) = 1 - e^{-kx}$ and the survival function is $P(X>x) = e^{-kx}$, where k is the rate parameter. Given P(X>5) = 0.35, we have: \[ e^{-5k} = 0.35 \] By definition of conditional probability: \[ P(X>10 \mid X>5) = \frac{P(X>10 \text{ and } X>5)}{P(X>5)} \] Since the event (X>10) is a subset of (X> 5), their intersection is just (X> 10). \[ P(X>10 \mid X>5) = \frac{P(X>10)}{P(X>5)} = \frac{e^{-10k}}{e^{-5k}} = e^{-10k + 5k} = e^{-5k} \] Since we already know $e^{-5k} = 0.35$, the answer is 0.35.
(Note: The provided answer in the source PDF, 0.353, appears to be a typo, as the mathematical derivation strictly leads to 0.35).
Step 4: Final Answer:
The conditional probability P(X> 10 | X> 5) is 0.35.