A lot of 25 light bulbs consists of N₁ defectives and N₂ working bulbs. The buyer accepts thelot of N₁ + N₂ = 25 bulbs if the number of defectives, X among n = 5 items taken at randomand without replacement from the lot is less than or equal to 1. The operating characteristiccurve is defined as the probability of accepting the lot, namelyOC (p) = P(X ≤ 1) = P(X = 0) + P(X = 1),where p = N₁/25 is the probability of a bulb being defective.(a) Show that the probability of the picking exactly X defectives follows a hypergeometricdistribution.(b) Determine the operating characteristic curve for p = 0.04, p = 0.08, p = 0.12 andp = 0.16.

Size of population There are 25 light bulbs out of which is the number of defective bulbs and…

Answered: A lot of 25 light bulbs consists of N₁…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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The PMF of the number of components K of a system that fails is defined by = {( PK (k)= ()(0.2)*(0.8) 4-k k= 0, 1,...,4 otherwise (a) What is the CDF of K? (b) What is the probability that fewer than two components of the system fail?
E. Alex is back on Tinder. Let X, be a random variable that is equal to 1 if i-th person is a match and to 0 otherwise. Suppose that each match happens independently of the others with probability 0.5. Let X1,..., X, denote a random sample. Denote Xn 2X, as %3D usual. (24) If n = 64, P(X <0.58) is approximately (a) 0.6 (b) 0.75 (c) 0.8 (d) 0.9
An environmental engineer collected 10 moss and 10 lichen specimens. The engineer instructs a laboratory intern to randomly select 15 of the specimens. The probability mass function of the number of lichen specimens selected at random is: a) H (x; 15; 10; 20) b) P (x; 10) c) Bnegative (x, 10, 0.666) d) B (x, 10, 0.666)
A lot of 25 light bulbs consists of N₁ defectives and N₂ working bulbs. The buyer accepts the lot of N₁ + N₂ = 25 bulbs if the number of defectives, X among n = 5 items taken at random and without replacement from the lot is less than or equal to 1. The operating characteristic curve is defined as the probability of accepting the lot, namely OC (p) = P(X ≤ 1) = P(X = 0) + P(X = 1), where p = N₁/25 is the probability of a bulb being defective. (a) Show that the probability of the picking exactly X defectives follows a hypergeometric distribution. (b) Determine the operating characteristic curve for p = 0.04, p = 0.08, p = 0.12 and p = 0.16. (c) Plot, using R, the operating characteristic curve for N₁ = 1 to N₁ = 4.
1.) The amount of time an air-conditioning technician to repair a unit is in between 1.9 and 5 hours which is found to be uniformly distributed. Let x be the time needed to fix an A/C unit. b.) Find the probability that a randomly selected A/C unit repair requires less than 3.5 hours?
Suppose (S, P) is a binomial distribution with n trials and probability of success p. Let X be the random variable X(k) = k³, where k is the number of successes. Calculate E[X].
B5. Let X₁, X₂, ..., Xn be IID random variable with common expectation µ and common variance o², and let X = (X₁ + + X₂)/n be the mean of these random variables. We will be considering the random variable S² given by (a) By writing or otherwise, show that S² (b) Hence or otherwise, show that n S² = (x₁ - x)². = Ĺ(X₂ i=1 X₁ X = (X₁-μ) - (x-μ) = Σ(X; -μ)² - n(X - μ)². i=1 ES² = (n-1)0². You may use facts about X from the notes provided you state them clearly. (You may find it helpful to recognise some expectations as definitional formulas for variances, where appropriate.) (c) At the beginning of this module, we defined the sample variance of the values x₁, x2,...,xn to be S = 1 n-1 n i=1 ((x₁ - x)². Explain one reason why we might consider it appropriate to use 1/(n-1) as the factor at the beginning of this expression, rather than simply 1/n. B6. (New) Roughly how many times should I toss a coin for there to be a 95% chance that between 49% and 510/ of my nain toon land Honda?
Let Y be the number of successes in n independent trials of a random experiment having probability of success . For n = 5, compute P(Y > 3).
4. An insurance company sells an automobile policy with a deductible of one unit. Let X be the amount of the loss having pmf 0.9 f(x) = x = 0 x = 1,2,3,4,5,6 where c is a constant. a) Find c. b) Find the expected value of the amount the insurance company must pay.
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(b) Test two independent circuits one after the other. On each test, the possible outcome is either overvoltage (0) or undervoltage (U). Assume that all circuits are overvoltage with probability 0.4. Let X be the number of overvoltage circuit in the first test and Y be the number of overvoltage circuit before you observed undervoltage. If both circuit is overvoltage or undervoltage, Y = 2. Find the joint PMF of X and Y.
3. Suppose it is known from large amounts of historical data that X, the number of cars that arrive at a specific intersection during a 20-second time period, is characterized by the following discrete probability function: 6x f(x) = e-6, for x = 0,1,2, ... a) Find the probability that in a specific 20-second time period, more than 8 cars arrive at the intersection.

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