Question

We say that a random variable is memoryless if P(X>x+y∣X>x)=P(X>y). Show that exponentially distributed random variables are memoryless. That is, show that this property holds if X∼Exp(λ)

Answer 1

Exponentially distributed random variables are memoryless, meaning that future probabilities do not depend on any past information. This property is proven by showing that the conditional probability P(X > x + k|X > x) is equal to P(X > k) for exponentially distributed random variables. Using the probability density function of the exponential distribution, we can derive the expression and simplify it to prove the memoryless property.

Step-by-step explanation:

The memoryless property states that for an exponential random variable X, the probability that X exceeds x+k, given that it has exceeded x, is the same as the probability that X would exceed k if we had no knowledge of it. In symbols, we can write this as P(X > x + k|X > x) = P(X > k).

To show that exponentially distributed random variables are memoryless, we can use the property of the exponential distribution that the probability density function is f(x) = me^(-mx). Now, let's prove the memoryless property:

1. Start with the left-hand side of the equation: P(X > x + k|X > x).
2. Using conditional probability, we can rewrite it as P(X > k + x)/P(X > x).
3. Now, substitute the exponential distribution's probability density function: (1 - F(k + x))/(1 - F(x)), where F(x) is the cumulative distribution function of X.
4. Using the expression for the cumulative distribution function of the exponential distribution, which is 1 - e^(-mx), we have (1 - e^(-m(k+x)))/(1 - e^(-mx)).
5. Simplify the expression using exponential laws, and we get e^(-mx - mk)/e^(-mx), which is equal to e^(-mk), or P(X > k).
6. Thus, the memoryless property holds for exponentially distributed random variables.