If X follows the Poisson probability law such that P (x 1) = P (x = 2). then ind the probability of 4 successes.
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- If a binomial experiment has probability p success, then the probability of failure is ____________________. The probability of getting exactly r successes in n trials of this experiment is C(_________, _________)p (1p)A hypothesis test produces a t statistic of t = +2.19. If the researcher is conducting a two-tailed hypothesis test with α = .05, how large does the sample have to be in order to reject the null hypothesis?A continuous random variable X is defined by: Solve: 1. f(x) = (3+x)²/16 ; -3≤ x≤ -1 2. f(x) = (6 - 2x)²/16 ; -1≤x≤1 3. f(x) = (3 - x²)/16 ; -1≤x≤3
- Show that (X+1)/(n+2) is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased?In a clinical study, a random sample of 540 participants agree to have their blood drawn, which is to be examined for the presence of antibodies against a certain contagious disease. It is found in 22% of the blood samples, which experimenters hope to extrapolate to the general population. From this random sample, 10 participants' blood samples are selected at random. If X is the number of samples out of the 10 who have these antibodies, what can we say about X? A. The sample size is not large enough for us to approximate X using a normal distribution B.The expected value of X is 22 C. X can be approximated using a normal distribution in lieu of a binomial distribution D. X has a sampling distribution that is normalQ)There are two tests for a disease, one is rapid and the other is slow. Given that an individual is infected, the rapid test will register positive 40% of the time, while the slow test will register positive 80% of the time; additionally, both tests will be positive 35% of the time. Assume currently, 20% of the population has the virus. Use Bayes theorem to calculate the chance that a person has the virus conditioned on getting negative results for both tests?
- Given three independent Bernoulli processes Xn, Yn, Zn with success probabilities given as 0.10, 0.87, and 0.26, respectively. X and Y are merged into one Bernoulli process which is merged with Z after. Find the resulting success probability of the last Bernoulli process. (Note: the answer should be in 4 decimal points)For conducting a two-tailed hypothesis test with a certain data set, using the smaller of n1-1 and n2-1 for the degrees of freedom results in df=11, and the corresponding critical values are t=+-2.201. Using the formula for the exact degrees of freedom results in df=19.063, and the corresponding critical values are t=+-2.093. How is using the critical values of t=+-2.201 more "conservative" than using the critical values of +- 2.093?If X1, X2, and X3 constitute a random sample of sizen = 3 from a Bernoulli population, show that Y =X1 + 2X2 + X3 is not a sufficient estimator of θ. (Hint:Consider special values of X1, X2, and X3.)
- Consider a bimominal experiment with 2 trials and p=0.4 d. Find the probability of at least one successIf X1, X2, ... , Xn constitute a random sample of size nfrom a geometric population, show that Y = X1 + X2 +···+ Xn is a sufficient estimator of the parameter θ.The lifespan X (years) of a species of pig has pdff (x) = 3x2e−x^3 , 0 < x < ∞.What is the probability that a pig will live at least 1 year? Given that it lives 1 year, what is the conditionalprobability that it will live for at least another year?