Let $T$ be exponential with parameter $λ.$ Let $X$ be discrete defined by $X=k,$ if $k≤T<k+1,$ $k=0,1,2,\dots$. Find the pdf of $X.$

Question

I am aware that this question has been asked already here, however there is no accepted answer to it yet. I have no idea where to start.

we know that

$$ f(t)= \lambda \times exp (-\lambda t)$$ then

$$ F(t)= 1 - exp (-\lambda t)$$

but i am not sure how to continue

Answer 1

Because $T$ is continuous, $$P(X = k) = P(k \le T < k+1) = F_T(k+1) - F_T(k)\\ = e^{-k\lambda} - e^{-(k-1)\lambda},$$ for $\lambda > 0$ and $k = 0, 1, \dots,$ where you have given $F_T$ in your Question.

The process you have in mind amounts to rounding exponential data down to the next lower integer; that is, taking the 'floor' function.

For exponential rate $\lambda = 1/5,$ one has $E(T) = 5$ and $V(T) = 25.$ The following computation in R evaluates $E(X) \approx 4.517$ and $Var(X) \approx 24.92$ by summing the first 100 terms in the relevant infinite series. An exact analytic evaluation is possible.

Also shown are the first ten probabilities. [In R, pexp is an exponential CDF.]

k = 0:100;  p = pexp(k+1,.2) - pexp(k,.2)
mu = sum(k*p); mu
[1] 4.516655       # aprx E(X) by summing early terms of series
vr = sum((k-mu)^2*p); vr
[1] 24.91682       # aprx Var(X) also from series
cbind(k, p)[0:10,]
      k          p
 [1,] 0 0.18126925
 [2,] 1 0.14841071
 [3,] 2 0.12150841
 [4,] 3 0.09948267
 [5,] 4 0.08144952
 [6,] 5 0.06668523
 [7,] 6 0.05459725
 [8,] 7 0.04470045
 [9,] 8 0.03659763
[10,] 9 0.02996360

A simulation of a million observations from the distribution of $X$ gives approximations of the above (mostly accurate to two or three significant digits).

set.seed(1205)
t = rexp(10^6, .2);  x = floor(t)
mean(x); var(x)
[1] 4.526318     # aprx E(X) by simulation
[1] 25.0892      # aprx Var(X) by simulation
plot(table(x)/10^6, pty="h")

table(x)/10^6
x
       0        1        2        3        4        5        6 
0.181598 0.148152 0.120972 0.099247 0.081509 0.066520 0.054763 
       7        8        9       10       11       12       13 
0.044482 0.036697 0.030249 0.024474 0.020415 0.016459 0.013413 
      14       15       16       17       18       19       20 
0.011077 0.008976 0.007304 0.006057 0.004946 0.004110 0.003317 
      21       22       23       24       25       26       27 
0.002796 0.002209 0.001832 0.001579 0.001220 0.001016 0.000856 
      28       29       30       31       32       33       34 
0.000709 0.000537 0.000461 0.000319 0.000327 0.000261 0.000209 
      35       36       37       38       39       40       41 
0.000148 0.000140 0.000106 0.000086 0.000083 0.000071 0.000060 
      42       43       44       45       46       47       48 
0.000037 0.000031 0.000036 0.000024 0.000019 0.000011 0.000015 
...