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i. Compute E[X (u, t)] for 0 ≤ t ≤ 2. Using your expression evaluate E[X(u, 1)] and E[X(u, 2)]. ii. Now as t becomes large, E[X(u, t)] μ (a constant). Find u (you may assume that N→ ∞ so that t can be arbitrarily large). Note that you can work this part independent of part (i). →

i. Compute E[X (u, t)] for 0 ≤ t ≤ 2. Using your expression evaluate E[X(u, 1)] and E[X(u, 2)]. ii. Now as t becomes large, E[X(u, t)] μ (a constant). Find u (you may assume that N→ ∞ so that t can be arbitrarily large). Note that you can work this part independent of part (i). →

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.EA: Extended Application Using Extrapolation To Predict Life Expectancy
Problem 7EA
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Problem 3. Means and Correlations. a. Here we consider mobile phone usage by Alice. Let us model her phone usage as follows: First we pick a random number S₁ uniformly dis- tributed between 0 and 1 minute. This represents the start time of the call. Alice uses the phone for exactly 1 minute each time so ends the first call at time E₁ = S₁ + 1. She then makes another call start- ing at a time S₂ where S₂ is uniformly distributed between E₁ and E₁ + 1 and ends the call at time E2 S2 + 1. This process contin- ues until Alice has made and completed N calls with Alice starting a call at a time Sk where Sk is uniformly distributed between Ek-1 and Ek-1 + 1 and ends the call at time Ek = Sk + 1, k = 1,2,..., N, with Eo 0. Let us define X(u, t) 0 when Alice is not using the phone and X(u, t) = 1 when Alice is using the phone, where u is a realization from an N-dimensional sample space, i.e., u = {U₁, U₂,...,UN}, with each u₁, i = 1,2,..., N, uniformly distributed in (Ei-1, Ei-1+1). Then X(u, t) is a random process. = = Below is an example of a sample function for N = 3. 1 0 S1 E1 S2 E2 S3 E3
Transcribed Image Text:Problem 3. Means and Correlations. a. Here we consider mobile phone usage by Alice. Let us model her phone usage as follows: First we pick a random number S₁ uniformly dis- tributed between 0 and 1 minute. This represents the start time of the call. Alice uses the phone for exactly 1 minute each time so ends the first call at time E₁ = S₁ + 1. She then makes another call start- ing at a time S₂ where S₂ is uniformly distributed between E₁ and E₁ + 1 and ends the call at time E2 S2 + 1. This process contin- ues until Alice has made and completed N calls with Alice starting a call at a time Sk where Sk is uniformly distributed between Ek-1 and Ek-1 + 1 and ends the call at time Ek = Sk + 1, k = 1,2,..., N, with Eo 0. Let us define X(u, t) 0 when Alice is not using the phone and X(u, t) = 1 when Alice is using the phone, where u is a realization from an N-dimensional sample space, i.e., u = {U₁, U₂,...,UN}, with each u₁, i = 1,2,..., N, uniformly distributed in (Ei-1, Ei-1+1). Then X(u, t) is a random process. = = Below is an example of a sample function for N = 3. 1 0 S1 E1 S2 E2 S3 E3
i. Compute E[X(u, t)] for 0 ≤ t ≤ 2. Using your expression evaluate E[X(u, 1)] and E[X(u, 2)]. ii. Now as t becomes large, E[X(u, t)] → µ (a constant). Find u (you may assume that N → ∞ so that t can be arbitrarily large). Note that you can work this part independent of part (i).
Transcribed Image Text:i. Compute E[X(u, t)] for 0 ≤ t ≤ 2. Using your expression evaluate E[X(u, 1)] and E[X(u, 2)]. ii. Now as t becomes large, E[X(u, t)] → µ (a constant). Find u (you may assume that N → ∞ so that t can be arbitrarily large). Note that you can work this part independent of part (i).
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