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- Let X1, X2, ..., Xn be independent and identically distributed random variables, X ∼ Exp(λ). Show that the sum X1 + X2 + · · · + Xn is Γ(n, λ) distributed.Machine X use X-rings to contain burning fuel in the combustion chamber. For aspecific model of machine X, it uses three X-rings (X1, X2, and X3). For a successfulcombustion experiment, there should be no fuel leakage and for that X1 must functionwith at least one of the other X-rings.Assuming that the X-rings behave independently of one another, that each X-ring hasa 0.9 probability of working. Construct a tree diagram to represent the information above.Using the mid- square method obtain the random variables using Z0= 7182 until the cycledegenerates to zero.
- For the random variables X,Y Cov(X,Y) = -0.9 if Z=3-X then what is Cov(Z,Y)=???Workers at a large toxic cleanup project are concerned that their white blood cell counts may have been reduced. Let x be a random variable that represents white blood cell count per cubic millimeter of whole blood in a healthy adult. Then μ = 7500 andσ ≈ 1750.† A random sample ofn = 70 workersfrom the toxic cleanup site were given a blood test that showedx = 6920.What is the probability that, for healthy adults,xwill be this low or lower?(a) How does the central limit theorem apply? Explain. The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈1750.0.The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈209.2. The central limit theorem does not apply because the sample size is too small.The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈25.0. (b) ComputeP(x ≤ 6920).(Round your answer to four decimal places.) P(x ≤ 6920)= (c) Based on your answer to…If X is a random variable, prove that Cov(X,X) = σX².
- Assume that a population of patients contains 30% of individuals who suffer from a certain fatal syndrome Z, which simultaneously makes it uncomfortable for them to take a life-prolonging drug X. Let Z = 1 and Z = 0 represent, respectively, the presence and absence of the syndrome, Y = 1 and Y = 0 represent death and survival, respectively, and X = 1 and X = 0 represent taking and not taking the drug. Assume that patients not carrying the syndrome, Z = 0, die with probability 0.5 if they take the drug and with probability 0.5 if they do not. Patients carrying the syndrome, Z = 1, on the other hand, die with probability 0.7 if they do not take the drug and with probability 0.3 if they do take the drug. Further, patients having the syndrome are more likely to avoid the drug, with probabilities p(X = 1|Z=0) = 0.9 and P(X = 1|Z = 1) = 0.6 . Based on this model, compute the joint distributions and for all values of x, y, and z. Present the following joint distributions in tables. [Hint:…A class contains 30 students. The probability of failing in one of the courses for each student is 0.02, but there is no chance to fail in more than one course. You know that a student failed in a course, the probability that the fail reason in non-attendance is 0.03. All other fail reasons are low average. The examinations are private, so the results are mutually independent. Let X and Y be the numbers of failed students due to non-attendance and due to low average, respectively, in that class this semester. Find the joint MGFSignal-to-Noise Ratio: If random variable X has mean μ ¹ 0 and standard deviation s > 0, the ratio r = |μ| /s is called the measurement signal-to-noise ratio (SNR) of X (sometimes SNR is defined as the square of this quantity.) The idea is that X can be expressed as X = μ+ (X − μ), with μ representing a deterministic constant-valued “signal”, and (X − μ) the random zero-mean “noise.” In applications, it is desirable to have an upper-bound, with a probabilistic guarantee, on the magnitude of the relative deviation of X from its mean μ, defined as D = |(X − μ) /μ| . That is, we are interested in finding a positive number a, such that P(D ≤ a) ≥ c where c is a pre-specified number representing our confidence in the upper-bound. If r = 10, provide an upper-bound a, with confidence level c = 0.95.
- In a Right Tailed Hypothesis test, the test statistic was found to be Z=2.74The rejection region included values greater than the critical value Zc=2.02 The conclusion would be to... Reject the null hypothesis because the test statistic is NOT in the rejection region Fail to reject the null hypothesis because the test statistic is in the rejection region Fail to reject the null hypothesis because the test statistic is NOT in the rejection region Reject the null hypothesis because the test statistic is in the rejection region Accept the null hypothesis because the test statistic is NOT in the rejection regionSuppose that X1, X2, X3, ..., X99, X100 are independent Bernoulli(1/2) random variables. What is E[(X1+X2+X3+...+X69+X70)*(X71 + X72 + ... + X99+X100)]?2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).