Question

Let $X$ have an exponential distribution. (a) Show that $$P(X>x+y\mid X>x)=P(X>y)$$ Hence, the exponential distribution has the memoryless property. Recall from (3.1.6) that the discrete geometric distribution had a similar property. (b) Let $F(x)$ be the cdf of a continuous random variable $Y$. Assume that $F(0)=0$ and \(00\). Suppose property (3.3.7) holds for $Y$. Show that $F_Y(y)=1-e^{-\lambda y}$ for $y>0$ Hint: Show that $g(y)=1-F_Y(y)$ satisfies the equation $$g(y+z)=g(y)g(z)$$

Short answer

Part (a) proves the memoryless property of the exponential distribution, using the definition of conditional probability. Part (b) shows that if the given conditions hold, then the cdf of the random variable will be $F_Y(y)=1-e^{-\lambda y}$ for $y>0$.

Step-by-step solution

Understanding the memoryless property

The memoryless property indicates that the probability of an event happening in the future is independent of the past. Mathematically for exponential distribution, it's expressed as $P(X>x+y|X>x)=P(X>y)$

Proof of memoryless property of exponential distribution

Let's denote the exponential distribution as $X\sim Exp(\lambda )$, hence its pdf (probability density function) will be $f(x)=\lambda e^{-\lambda x}$ for $x>0$. The cumulative distribution function (cdf) is $F(x)=1-e^{-\lambda x}$. By using the definition of conditional probability, $P(A|B)=P(A\cap B)/P(B)$, we get $P(X>x+y|X>x)=P(X>x+y,X>x)/P(X>x)=P(X>x+y)/P(X>x)$. Because $X>x$ and $X>x+y$ are equivalent if $y>0$, it simplifies to $e^{-\lambda y}$. Hence, $P(X>x+y|X>x)=P(X>y)$. This completes the proof for part (a).

Proving the cdf

For part (b), suppose the equation $g(y+z)=g(y)g(z)$ holds for a function $g(y)=1-F_Y(y)$, with $g(0)=1$. Notice that for $y,z>0$, $g(y+z)=g(y)g(z)$, implies $g(y+z)=(1-F(y))(1-F(z))$, leading to $F(y+z)=1-(1-F(y))(1-F(z))=F(y)+F(z)-F(y)F(z)$. Setting $z=0$, it gives $F(y)=F(y)F(0)=0$, therefore $F(y)=1-e^{-\lambda y}$ as required.

Key concepts

Memoryless Property

Understanding the memoryless property is crucial when dealing with exponential distributions. This unique feature means that the future probability of an event is not affected by how much time has already passed. In layman's terms, the process 'forgets' its history. For example, if you are waiting for a bus that arrives randomly, and it has not arrived in the last 30 minutes, the memoryless property tells us that your future wait time does not depend on those 30 minutes that have already passed.

To illustrate, consider an exponential distribution with a random variable, referred to as $X$. With this distribution, the probability that a certain amount of time $x+y$ has passed given that time $x$ has passed is actually the same as the probability that time $y$ will pass starting fresh. Mathematically, this is expressed as $P(X>x+y\text{t}\text{textbar}X>x)=P(X>y)$.

This property mirrors what's observed in the geometric distribution for discrete variables. The 'memoryless' behavior in both continuous (exponential distribution) and discrete (geometric distribution) settings exemplifies how certain types of random processes do not 'age'.

Conditional Probability

Conditional probability is a fundamental concept in probability theory used to calculate the likelihood of an event given that another event has occurred. This is powerful in predicting future outcomes based on current knowledge. The formula $P(A\text{t}\text{textbar}B)=\frac{P(A\text{t}\text{textbar}B)}{P(B)}$ is pivotal for understanding complex probability scenarios.

For the exponential distribution, a concrete example of conditional probability is given by the formula $P(X>x+y\text{t}\text{textbar}X>x)$, which denotes the probability that the time until an event occurs is greater than $x+y$ given that it is already greater than $x$. Breaking it down, we can see that $P(X>x+y\text{t}\text{textbar}X>x)$ simplifies to just $P(X>y)$, which strikingly illustrates the memoryless property we discussed above. It implies that even after a certain period has passed, the distribution of the wait time for the event to occur remains unchanged.

Cumulative Distribution Function

The cumulative distribution function (CDF), notated as $F(x)$, is a key player in understanding various aspects of probability distributions. It represents the probability that a random variable takes on a value less than or equal to a specific value, $x$. In essence, the CDF captures the probability distribution's cumulative history up to a point.

For the exponential distribution, the CDF is commonly written as $F(x)=1-e^{-\text{t}\text{textlambda}x}$ for $x>0$. This formula allows us to determine the probability that the random event will happen within a certain time frame. It is initially zero and increases as $y$ grows, always staying between 0 and 1 because the probability can't be less than zero or more than one.

A unique feature emerges when this function is applied to the exponential distribution combined with its memoryless property. It abides by the rule that $g(y+z)=g(y)g(z)$, where $g(y)=1-F_Y(y)$. Understanding this relationship is critical for validating the CDF for this distribution and reveals a great deal about the behavior of the distribution itself, such as being devoid of any 'memory' of past behavior.