6. We have three light bulbs with lifetimes \( T_1, T_2, T_3 \) distributed according to Exponential(\(\lambda_1\)), Exponential(\(\lambda_2\)), Exponential(\(\lambda_3\)). In other words, for example bulb #1 will break at a random time \( T_1 \), where the distribution of this time \( T_1 \) is Exponential(\(\lambda_1\)). The three bulbs break independently of each other. The three light bulbs are arranged in series, one after the other, along a circuit—this means that as soon as one or more light bulbs fail, the circuit will break. Let \( T \) be the lifetime of the circuit—that is, the time until the circuit breaks. --- **(a)** What is the CDF of \( T \), the lifetime of the circuit? **(b)** Next, suppose that we only check on the circuit once every second (assume the times \( T_1, T_2, T_3, T \) are measured in seconds). Let \( S \) be the first time we check the circuit and see that it’s broken. For example, if the circuit breaks after 3.55 seconds, we will only observe this when 4 seconds have passed, and so \( S = 4 \). Calculate the PMF of \( S \). **(c)** Finally, suppose that instead of checking on the circuit every second, we instead do the following: after each second, we randomly decide whether to check on the circuit or not. With probability \( p \) we check, and with probability \( 1 - p \) we do not check. This decision is made independently at each time. Now let \( N \) be the number of times we check and see the circuit working. For example, if the circuit breaks at time 3.55, and our choices were to check at time 1 second, not to check at times 2 or 3 or 4, and to check at time 5, then \( N = 1 \), since the circuit was broken the 2nd time we checked. What is the PMF of \( N \)? (Hint: start by finding the joint PMF of \( N \) and \( S \). It's fine if your answer is in summation form.)

A doctor prescribes 100 milligrams of a therapeutic drug that decays by about 20% eachhour (ie. after 1 hour, 20% of the drug is gone). Assume the quantity of the drug is modeled by anexponential equation. What is the half-life of the drug? Please use a calculator to compute the finalanswer and round down to two decimal places.

Determine whether the following scenarios are linear or exponential. As summer approaches, I noticed that for every increase in 5 degrees, the number of people I see at the pool increases by about 3. The amount of Francium,a radioactive element, decays by about 3.1% every minute. The number of flu cases increases by about 1.6% per month during flu season. Based on Mrs. Olivet's Test 4 results, students tended to score about 4 more points on their test for every hour they studied. The percentage of the population that has been vaccinated for COVID-19 increased by about 0.25 percentage points every day in January. The population of bugs in Monica's walls decreased by 80% every week while the exterminator was treating her home. During a rainstorm the reading from the rain gauge increased by a quarter inch every hour. Selene didn't want her parents to notice she was stealing candy from the jar, so every day she only ate 1/8 of what was left. While writing an essay, Carson writes 7…

A app.edulastic.com lever | Teacher.. Launch Meeting -. C Classes A 8 Advisory Michi. O Launch Meeting. E Edulastic: Format. Question 20/26 > NEXT A BOOKMARK A population of bacteria, B , can be modeled using the exponential function B (x) = 3 (2)" , where a is the number of days since th population was first observed. Which domain is most appropriate to use to determine the population over the course of the first weel- A00 15 tv MacBook Air 80 888 DII DD F1 F3 F4 F5 F6 F7 FB @ $ % & 1 2 3 4 8 Q W E R T Y S K