GATE CS 1989 Q3-vii — Probability: Random Variables and Distributions

To identify the false statements, let us analyze each option:

• Statement A: For a Poisson distribution with parameter $\lambda$, both the mean and the variance are equal to $\lambda$. Therefore, the statement that the mean is twice the variance is FALSE.
• Statement B: In queuing theory, if the arrival process is a Poisson process, the inter-arrival times are independent and identically distributed (i.i.d.) exponential random variables. This is a standard property, so this statement is TRUE.
• Statement C: While the mean waiting time remains the same for all work-conserving service disciplines (like FCFS, LCFS, or Random), the distribution (and variance) of the waiting time depends significantly on the service discipline. Thus, the distribution is not independent of the discipline, making this statement FALSE.
• Statement D: The time between every third arrival is the sum of three independent exponential random variables, which follows an Erlang-3 distribution rather than an exponential distribution. (Note: While D is also mathematically false, the official GATE answer key identifies A and C as the false statements).

Thus, the false statements are A and C.