Q.6.24 - A First Course in Probability

Question

If X is exponential with rate λ, find P{[X] = n, X − [X] … x}, where [x] is defined as the largest integer less than or equal to x. Can you conclude that [X] and X − [X] are independent? 

Step-by-Step Solution

Answer

The required probability is $e^{-\lambda n}(1-e^{-\lambda n})$, but these random variables are not independent. 

Step 1: Content Introduction

Observe that random variable  $X-\left[X\right]$ describes the decimal remainder between the true value of X and its largest integer approximation. Hence $X- \left[X\right]\in [0,1]$.

Step 2: Content Explanation

The event  means that $X\in [n,n+x]$, so the probability for that event is, 

$$\begin{gathered} P\left(\right[X]=n,X -[X]\le x)=P(X\in [n,n+x\left]\right) \\ =F_x(n+x)-F_x\left(n\right) \\ =e^{-\lambda n}-e^{-\lambda (n+x)} \\ =e^{-\lambda n}(1-e^{-\lambda x}) \end{gathered}$$

Also, these random variables are not independent. Observe that,

$$\begin{gathered} P\left(\right[X]=n)=F_x(n+1) \\ =e^{-\lambda n}-e^{-\lambda (n+1)} \\ =e^{-\lambda n}(1-e^{-\lambda }) \end{gathered}$$

and because of the memoryless properties of exponential distribution we see that

$$\begin{gathered} P(X-[X]\le x)=F_x\left(x\right) \\ =1-e^{-\lambda x} \end{gathered}$$