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- The random variable X has a Bernoulli distribution with parameter p. A random sampleX1, X2, . . . , Xn of size n is taken of X. Show that the sample proportionX1 + X2 + · · · + Xnnis a minimum variance unbiased estimator of p.Given that X1, X2, . . . , Xn forms a random sample of size n from a geometric population withparameter p, show thatY =n∑j=1Suppose X and Y are continuous random variables such that the pdf is f(x,y)=x+y with 0≤x≤1,0≤y≤1. a)Draw a graph that illustrates the domain of this pdf. b)Find the marginal pdfs of X and Y.c)Compute ux,ux,var x,var y, cov(x,y) ,and pd (Rho d)) Determine the equation of the least squares regression line and draw it on your graph.Let X 1 ,X 2 ,...,X n be a r.s. of size n from a distribution N(theta, 1) find best critical region of size a for testing H 0 : theta=0 against H 1 : theta=1 , also find the value of c if n = 25 and a = 5%.
- The table shows data from a stadium concession stand, representing average temperature (x) in degrees Fahrenheit and the number of popsicles sold (y). A 4-column graph with 6 rows. Column 1 is labeled x with entries 65, 72, 85, 88, 92, sigma-summation x = 402. Column 2 is labeled y with entries 22, 25, 52, 65, 78, sigma-summation y = 242. Column 3 is labeled x squared with entries 4,225, 5,184, 7,225, 7,744, 8,464, sigma-summation x squared = 32,842. Column 4 is labeled x y with entries 1,430, 1,800, 4,420, 5,720, 7,176, sigma-summation x y = 20,546. Which regression equation correctly models the data? y = 0.60x – 17 y = 0.60x – 0.16 y = 2.09x – 114 y = 2.09x – 120Estimate the average value of f(x,y)=xy over the rectangle R=[−7,5]×[0,2] using a Riemann sum with m=4 and n=2. Take the sample values to be upper right corners. Select one: a. -4 b. -15/2 c. 3/4 d. -2 e. 5/2Estimate the area under the graph of f(x)=ln(x) over the interval [1,2] using five approximating rectangles and the sample points x_i* {1.14, 1.38, 1.43, 1.64, 1.93}
- 7. Assume that X₁,..., Xn is a random sample from a Bernoulli (p) and let Yn n 1/1 X₁. For p # 1/2, Determine the asymptotic distribution of √n[Yn(1 – Yn) - p(1 - p)]. 7. Assume that \( X_1, ..., X_n \) is a random sample from a Bernoulli(p) and let \( {Y}_n = \1/n\sum_{i=1}^{n} X_i \). For \( p =/ 1/2), Determine the asymptotic distribution of \( \sqrt{n}({Y}_n(1 - {Y}_n) - p(1 - p)) \).A raptured pipe in a North Sea oil rig produces a circlar oil slick that is y meters thick at a distance x meters from the rapture. Turbulance makes it difficult to directly measure the the thickness of the slick at the source (where x=0), but for x >0, it is found that y= 0.5(x2+3x)/x3+x2+4x. Assuming the oil slick is continiously distributed, how thick would you expect it to be at the source?Use the z-table, find the critical region/s for each: a) a = 0.015, two-tailed test b) a = 0.025, left-tailed test c) a = 0.0125, right-tailed test
- 5 We randomly select a point (X,Y) from the triangle {(x,) : x,y \in {1,2,...,6}, y <= x} in the following nonuniform way. "First, select X discrete uniformly from {1,2,...,6}. Then, given X=x, select Y discrete uniformly from {1,...,x}. " a) Find the joint pmf of (X,Y) b) Find the marginal pmfs of X and Y c) Find the conditional pmf of X given Y = 2. d) Write R-code that simulates 10^4 pairs (X,Y) and use the table() command to to check if your results of a) b) and c) are correct.Suppose Y1,Y2,··· ,Yn are i.i.d. continuous uniform(0,1) distributed.(a) Prove that the kth-order statistic, Y(k), has Beta distribution with α = k and β = n−k+ 1.(b) What is the distribution of the median of Y1,Y2,··· ,Yn when n is an odd integer?(c) Find the joint density of the middle two of Y(1),Y(2),··· ,Y(n) when n is an even integerAssume the discrete population model: Nt+1=bNt, t=0,1,2,... Assume that the population increases by 2% each generation. a) Determine b b) Find the size of the puopulation at generation 10 when N0=20 c) After how many generations will the population have doubled?