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.(c) Plot, using R, the operating characteristic curve for N₁=1 to N₁ = 4.

Let X1, X2, X3, X4, X5 be random variables defined as follows: Xi = 1 if the ith item chosen is…

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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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?
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
9. This is a simplified inventory problem. Suppose that it costs c dollars to stock anitem and that the item sells for s dollars. Suppose that the number of items thatwill be asked for by customers is a random variable with the frequency functionp(k). Find a rule for the number of items that should be stocked in order tomaximize the expected income. (Hint: Consider the difference of successiveterms.)
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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.
2. Let X1, X2,..., X36 be a random sample of size 36 from the geometric distribution with p.m.f. f(x) = (1/4)"-'(3/4), x = 1, 2, 3, .... Approximate the following probabilities. (Hint, E(X) = , Var(X) = ) 36 (a) P( 46 < X; < 49 i=1 (b) P(1.25 < X < 1.50)
6. Let X1, X2, ...X100 denote a random sample of size 100 from an exponential distribution with mean 1. Let Y equal the sum of all the X's. Approximate the probability that Y is greater than 110. |
A child gets a weekly allowance (a certain non-negative number of dollars). The amount (in dollars) is random with expectation equal to 6 and variance equal to 6. Let’s denote this random variable by X. I want to estimate (from above) the chances that the child gets 10 dollars or more on a given week. In other words, I want to find c such that P(X ≥ 10) ≤ c, with c that is as small as possible.(a) Use the Markov inequality to find c such that P(X ≥ 10) ≤ c.(b) Use the Chebyshev inequality for the variance to find c such that P(X ≥ 10) ≤ c.(c) Read about the one-sided Chebyshev inequality, and use it to find c such that P(X ≥ 10) ≤ c.
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