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- Given (x + 2y) ,0With an American penny, the likelihood of getting H when it is spun on edge is 0.3. If X is the random variable where X(H ) = 1, X(T ) = −1, find the expected value E(X), the variance, Var(X), and express X in its standard form.1. Let X be an RV with PDF a. Find the mean of X. b. Find the variance of X. f(x) = {¹ -=| - |x| for x < 1 otherwised. Denote the expected return and its standard deviation as functions of π byμ(π ) and σ (π ). The pair (μ(π ), σ (π )) trace out a curve in the plane as πvaries from 0 to 1. Plot this curve. I need to draw a graph from this??A technician discovered that the cumulative distribution function (CDF) of the lifespan of bulb in years is given by f(y) = -10 ye 10 100 0Let X be a normal random variable with mean 85 and a variance of 25 (i.e., X ∼ N (85, 25)). step1: Let M = aX + b for some constants a, b not equal to 0. Write an expression for the pdf of M . step 2: Suppose that X represents an approximate distribution of the final scores in a certain math course (ignore the fact that this approximation can technically have scores that are greater than 100 or less than 0). If the teacher were to curve the scores, it means he would determine a function to apply to the scores to achieve a desired distribution (assume an affine function in this case, as in the previous part). Suppose he wants the scores to be normally distributed with mean 80 and a standard deviation of 4. What should he choose for a and b? What students would see their score lowered, and what students would see their score increased?Find the two-tailed U-table value when n1 = 5, n2 = 6, and alpha = 0.05.The next four questions (29 to 32) refer to the following: To most Canadians, earthquakes are viewed as rare occurrences, but they are actually quite common. In just one month in 2001, Natural Resources Canada recorded 215 earthquakes that affected Canada from B.C. to Nunavut to Newfoundland. We would like to determine whether there is a linear relationship between the location of an earth- quake X (measured in degrees latitude north of the equator) and the magnitude of the earthquake Y (measured on the Richter scale). The explanatory and response variables are measured for a sample of 13 earthquakes. The equation of the least squares regression line is ŷ = -3.05+ 0.10x. The ANOVA table is showm below: ге Source of Variation df Sum of Squares Mean Square F Regression Error Total 1.40 25.10Suppose that y, is an independent response variable (i = 1.....n) with mean, such that g(i) = n₁ = X₁B, where g(.) is a link function, X, is a vector of covariates and 3 is a px 1 vector of coefficients. Let the variance of y; be Var (Y) = V(i), where V(.) is a known function and is a scale parameter. Given: Zi= g'(pi) (Yi - Pi) + Ni w₁ = [V (pi) (g'(pi))²]-¹ 8 where g'(μ) = 9(μ) (a) Show that E(Z₁) = X₂B (b) Show that Var (Z₁) = w¹o. (c) Estimate 3 by minimizing Σ, wi( E(Z₁))² with respect to 3. After you found 8, find the Var(8) and E(3). What do you conclude?Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON