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

The correct answer is 0.35.

Step-by-step Explanation:

An exponential random variable X possesses a fundamental property known as the memoryless property. Mathematically, for any real numbers s, t > 0, the memoryless property of an exponential distribution is expressed as:

$$P(X>s+t|X>s)=P(X>t)$$

This property states that the probability of the event occurring after an additional time t, given that it has already survived up to time s, is the same as the initial probability of it surviving past time t. In other words, the history up to time s is forgotten.

We are asked to find the conditional probability:

$$P(X>10|X>5)$$

We can rewrite the values in the conditional probability expression to match the memoryless property formula. Let s = 5. Then, we can express 10 as s + t, which means 10 = 5 + 5. Thus, we have t = 5.

Applying these values to the memoryless property equation:

$$P(X>10|X>5)=P(X>5+5|X>5)=P(X>5)$$

We are given that:

$$P(X>5)=0.35$$

Therefore, substituting this value back into our equation, we obtain:

$$P(X>10|X>5)=0.35$$