Suppose that X1 and X2 are independent and exponentially distributed random vari- Let Y1 = X1 + X2 and Y2 X1 Find the joint pdf of ables, both with mean X2 Yı and Y2 (including its domain).
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![Suppose that X1 and X2 are independent and exponentially distributed random vari-
1
ables, both with mean . Let Y1 = X1 + X2 and Y2
X1
Find the joint pdf of
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Y1 and Y2 (including its domain).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64e6d4e1-4ef0-4619-aca9-13ba7dc48233%2F58a6471f-9832-44d7-ad9b-caa9a518d661%2Fi7fa6w6_processed.png&w=3840&q=75)
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- Suppose that Y₁, Y₂, ..., Ym is a random sample of size m from Gamma (a = 3, B = 0), where 0 is not known. Check whether or not the maximum likelihood estimator Ô is a minimum variance unbiased estimator of the parameter 8.Suppose that a random man's weight from a certain population is M~ Normal (μ = 200, o²49) and a random woman's weight is W ~ Normal( 140, o² = 25). If M and W are independent, what is the variance of the difference between a man's weight and a woman's weight, i.e. Var (M-W)? (a) 340 (b) 24 (c) 140 (d) 200 (e) None of the aboveSuppose that f (x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.
- Suppose that X₁, X₂, Xn and Y₁, Y2, . Yn are independent random samples from populations with means ₁ and ₂ and variances of and o2, respectively. Show that X - Y is a consistent estimator of μ₁ - 2.Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.Let the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.
- Suppose X~ Uniform [10, 50], find: P70 Variance (X)Suppose model (XY, XZ, YZ) holds in a 2 x 2 x 2 table, and the common XY conditional log odds ratio at the two levels of Z is positive If the XY and YZ conditional log odds ratios are both positive or both negative, show that the XY marginal odds ratio is larger than the XY conditional odds ratio.2 If X₁ and X₂ are the means of independent ran- dom samples of sizes n₁ and n₂ from a normal population with the mean u and the variance o2, show that the vari- ance of the unbiased estimator 2 w X₁ + (1-w). X₂ is a minimum when @ = n1 n₁ + n₂ idnu
- Suppose a random variable X has the following pdf. The random variable X has a Beta distribution with α = 3 and β = 2. Suppose we wanted to determine Var( 10X - 3 ). By the rules for variances, Var( 10X - 3 ) = * Var(X) + where, using the fact that X has a Beta distribution, Var(X) = Express answer as a decimal. Do not round.An electrochemical engineer has manufactured a new type of fuel cell (a type of battery)which has to undergo testing to prove its duration: the time it takes to go from fullycharged to completely uncharged, under a fixed nominal load. From the computational simulation models she has, the variance of the duration is σ2 = 4 h2 (hours squared)but she wants to estimate the mean duration time μ. To achieve this she is determinedto do the tests multiple times in independent but identical conditions. Can you findwhat is the smallest number of these tests that she has to do in order for her estimatedmean duration to be within ±0.2 h tolerance of the true mean with 95% certaintyLet x1, x2, ..., n represent a random sample from a distribution with mean E(x) and variance Var(x). Show that Cov(x, x₁ - x) = 0.
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