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- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0If a random variable X has a discrete uniform distribution. fx(x)=1/k for x=1,2,..,k;0 otherwise. Derive P.G.F of X and compute E(2x+1)Let X1 and X2 be observations of a random sample of size n = 2 from a Cauchy Distribution.Find P(X1 < −1 and 1 < X2)
- In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Poisson distribution with probability mass function f(x), where f(x) = θ x e −θ/x! and θ is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was.Which of the following is NOT a condition that ALWAYS has to be satisfied to apply the Central Limit Theorem to the sample mean? a) The sample size must be larger than 30 b) There are no obvious outliers in the sample if the sample size is less than 30 c) There are no extreme outliers in the sample if the sample size is at least 30. d) The sample observations are independent.Suppose that you enter a fantasy baseball league. Suppose that the 2021 team budget, say , is randomly drawn from a uniform distribution on the interval , where the unit is U.S. million dollars. In addition, suppose that after the value has been observed , the 2022 team budget, say , is randomly drawn from a uniform distribution on the interval . In other words, the 2022 budget is at most as large as the 2021 budget. a) For any given value of x(50<x<350), obtain E[Y|X=x] b) In view of part (a), obtain E[Y|X] c) Atlanta Braves won the 2021 World Series title. Their estimated 2022 payroll is about $130 million. Would your 2022 fantasy baseball budget be on average larger than their 2022 payroll? Explain briefly.
- Which of the following statements about the Central Limit Theorem (CLT) is correct?1. The CLT states that the sample mean is always equal to the population mean.2. The CLT states that the sampling distribution of the sample mean is approximately normal for large sample sizes (n > 30).3. The CLT states that the sample mean is equal to the population mean, provided that n > 30.4. The CLT states that the sampling distribution of the population mean is approximately normal, provided that n > 30. In a distribution plot, if the mean is equal to the median, is the possible shape of the distribution skewed to the right, skewed to the left, or symmetric? If batteries have lives that are normally distribute with mean 2, 800 hours and standard deviation 400 hours, using the 68-95-99.7 rule to approximate the percentage of batteries having a life less than 3, 600 hours.they take samples of 4 fireworks for quality co from and examine them for defects. let X be the number of defective fireworks in the sample of 4.According to the Central Limit Theorem, the distribution of sample means is NOT normal if _________________. A) The population from which the samples are selected is not normally distributed and the sample size is small (n < 30). B) The population from which the samples are selected is not normally distributed and the sample size is large (n >= 30). C) The population from which the samples are selected is normally distributed and the sample size is small (n < 30). D) The population from which the samples are selected is normally distributed and the sample size is large (n >= 30).
- Based on the information above, are the assumptions met? Based on the information above, are the assumptions met? No, because there is a simple random sample but the population is not normally distributed Yes, because there is a simple random sample and the sample size is large enough for the Central Limit Theorem Yes, because there is a simple random sample Yes, because there is a simple random sample and the population is normally distributedIf 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 Central Limit Theorem says A. When n>30n>30, the sampling distribution of x¯¯¯x¯ will be approximately a normal distribution.B. When n<30n<30, the original population will be approximately a normal distribution.C. When n>30n>30, the original population will be approximately a normal distribution.D. When n<30n<30, the sampling distribution of x¯¯¯x¯ will be approximately a normal distribution.E. None of the above