Poisson and Birth-death Processes - Stochastic Processes - Statistics Previous Year Questions

1

2019 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2019

Suppose customers arrive at an ATM facility according to a Poisson process with rate 5 customers per hour. The probability (rounded off to two decimal places) that no customer arrives at the ATM facility from 1.00 pm to 1.18 pm is ...

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2

2020 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2020

Let \(\{N(t), \ t \geq 0\}\) be a Poisson process with rate \(\lambda = 2\). Given that \(N(3) = 1\), the expected arrival time of the first event of the process is

A

\(1\)

B

\(\frac{3}{2}\)

C

\(\frac{2}{3}\)

D

\(3\)

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3

2020 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2020

In a pure birth process with birth rates $\lambda_n = 2^n, n \geq 0$, let the random variable $T$ denote the time taken for the population size to grow from 0 to 5. If Var($T$) denotes the variance of the random variable $T$, then $256 \times$ Var($T$) = ______________

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4

2021 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2021

Let customers arrive at a departmental store according to a Poisson process with rate 10. Further, suppose that each arriving customer is either a male or a female with probability 1/2 each, independent of all other arrivals. Let N(t) denote the total number of customers who have arrived by time t. Then which one of the following statements is NOT true?

A

If S_2 denotes the time of arrival of the second female customer, then P(S_2 ≤ 1) = 25 ∫_0^1 s e^{-5s} ds

B

If M(t) denotes the number of male customers who have arrived by time t, then P(M(1/5) = 0 | M(1) = 1) = 4/5

C

E[(N(t))^2] = 100t^2 + 10t

D

E[N(t)N(2t)] = 200t^2 + 10t

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5

2021 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2021

Consider an amusement park where visitors are arriving according to a Poisson process with rate 1. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have common probability density function
\(f(x) = \begin{cases} e^{-x}, & x > 0, \\ 0, & \text{otherwise}. \end{cases}\).
If at a given time point, there are 10 visitors in the park and \(p\) is the probability that there will be exactly two more arrivals before the next departure, then \(\frac{1}{p}\) equals __________

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6

2022 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2022

Consider a Poisson process \(\{X(t), t \ge 0\}\). The probability mass function of \(X(t)\) is given by \(f(t) = \frac{e^{-4t} (4t)^n}{n!}, \quad n = 0, 1, 2, ...\). If \(C(t_1, t_2)\) is the covariance function of the Poisson process, then the value of \(C(5, 3)\) (in integer) is equal to ______

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7

2023 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2023

Let {N(t)}_{t≥0} be a Poisson process with rate 1. Consider the following statements.
(I) \( P(N(3) = 3 \mid N(5) = 5) = \binom{5}{3} \left(\frac{3}{5}\right)^3 \left(\frac{2}{5}\right)^2 \).
(II) If \( S_5 \) denotes the time of occurrence of the \( 5^{th} \) event for the above Poisson process, then \( E(S_5 \mid N(5) = 3) = 7 \).
Which of the above statements is/are true?

A

Only (I)

B

Only (II)

C

Both (I) and (II)

D

Neither (I) nor (II)

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8

2024 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2024

Let \(\{N(t)\}_{t \geq 0}\) be a Poisson process with rate \(\lambda\), where \(\lambda > 0\) is an unknown parameter. Staring from the origin, an intercity road has \(N(t)\) number of potholes up to a distance of \(t\) kilometers. Staring from the origin, potholes are found at the following distances (in kilometers) \(0.9, 1.3, 1.8, 2.7, 3.4, 4.1, 4.7, 5.5, 6.2, 6.8, 7.4, 8.1, 8.9, 9.2, 9.7\). Based on the above data, the method of moment estimate of \(\lambda\) equals ______________ (rounded off to two decimal places).

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9

2025 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2025

Let {N(t); t ≥ 0} be a homogenous Poisson process with the intensity/rate λ = 2. Let
X = N(6) - N(1)
Y = N(5) - N(3)
W = N(6) - N(5)
Z = N(3) - N(1)
Then which one of the following options is correct?

A

Cov(W, Z) = 2

B

Y + Z ~ Poisson(10)

C

Pr(Y = Z) = 1

D

Cov(X, Y) = 4

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10

2026 · Statistics · Stochastic Processes · Poisson and Birth-death Processes

Statistics (ST) 2026

Let $\{N(t); t \geq 0\}$ be a homogeneous Poisson process with rate 3, and let $T_1$ denote the first arrival time. Then which of the following statements is/are correct?

A

$E(T_1) = 3$

B

$E(T_1) = \frac{1}{3}$

C

$E(T_1 | N(2) = 4) = \frac{6}{5}$

D

$E(T_1 | N(3) = 4) = \frac{2}{5}$

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