~x(n) (32) i=1 Exercise 11. Suppose X1, X2,.,Xn is an i.i.d. sample of observations of a random variable X ~ N(u, o2), and X is the sample mean. Show that (X – µ)² ~x°(1) (33) o2 Take care to give a complete explanation, using only those results that have been established! 14
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- Please help me for Question 13. Consider a pdf f so that a random variable X~ f has expected value E[X] = 3 and variance V[X] = 10. Now consider n = 10 i.i.d. random variables X1, X2,..., X10 drawn from f. Let X = 10 (a) What is E[X]? (b) What is VIX]? (c) What is the standard deviation of X? (d) Which is larger PX > 4] or P[X > 4]? 10 ₁=1 Xi.2. Let Y₁, Y₂, Y3, Y4, Y5 and X₁, X₂, ..., X, be independent and normally distributed random samples from populations with means μ₁ = 2 and μ₂ = 8 and variances ₁² = 5 and ₂² = k₁ respectively. Suppose that P(X-Ỹ > 10) = 0.02275, find the value of 0₂² = k. (5) Suppose that Y. Y y and X. X. X are independent normally distributed random
- If we let X1, X2,...Xn be a random sample from the distribution N( μx = 8, σx = 4), then what would the distribution of (X-8)/(4/√n) be? If we let X1, X2,...X10 be a random sample from the distribution N( μ, σ), then what is the value k so that P( (X-μ)/(σ/√10) < k ) = .05?(e) Show that the sample mean of a simple random sample (drawn without replacement from a normal population) is dis- tributed normally with mean u and variance -² (1--=-1). n where N and n are population and sample sizes respectively. 4 In the 1 1 TI9. (8') Let X,, X,, X, be a sample of size n=3 from a distribution with unknown mean u, -0 < u<0, where the variance o is a known positive number. Show that both à =i and ê, = 2X, + X, +5X, are unbiased estimators for H.
- Find the first two moments about the mean for the random variable X defined by X = -2 with prob. 1/3 3 with prob. 1/2 = 1 with prob. 1/6. %3DIf we let X1, X2,...X10 be a random sample from the distribution N( μ, σ), then what is the value k so that: P( (X-μ)/(σ/√10) < k ) = .05 P( (X-μ)/(S/√10) < k ) = .05 P( k < (X-μ)/(S/√10) ) = .05Let Y, represent the ith normal population with unknown mean #, and unknown variance of for i=1,2. Consider independent random samples, Ya, Ya, Yin, of size n,, from the ith population with sample mean Y, and sample variance S² = (₁-P) ². (d) Define V₁, a function of S1, that has a chi-square distribution. (e) Is V₁ a pivotal function to find a confidence interval for o?? Explain with argument. (f) Find a (1-a) x 100% confidence interval for of