Problem 1: The density function of a continuous random variable X is shown below. Find (a) the mode. (b) the median, and (c) the mean. f(x)= [0.9391 sin(√2-1) for 0.8 < x < 2.2 To elsewhere
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Q: shown below. Find (a) the mode. (b) the median, and (c) the mean. 0.9391 sin(√2x - 1) for 0.8 <x<2.2…
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- 1. Let X be a random variable having pdf f(x) = 6x(1 – x) for 0 < I < 1 and 0 elsewhere. Compute the mean and variance of X. 2. Let X1, X2,..., X, be independent random variables having the same distribution as the variable from problem 1, and let X, = (X1+ ·.+Xn). Part a: Compute the mean and variance of X, (your answer will depend on n). Part b: If I didn't assume the variables were independent, would the calculation in part a still work? Or would at least part of it still work? 3. Suppose that X and Y are both independent variables, and that each has mean 2 and variance 3. Compute the mean and variance of XY (for the variance, you may want to start by computing E(X²Y²)). 4. Suppose that (X,Y) is a point which is equally likely to be any of {(0, 1), (3,0), (6, 1), (3, 2)} (meaning, for example, that P(X = 0 and Y = 1) = }). Part a: Show that E(XY) = E(X)E(Y). Part b: Are X and Y independent? Explain. 5. Let X be a random variable having a pdf given by S(2) = 2e-2" for 0EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.6. Find the mean and variance of fx (x) = 2e eA* la.2...) (x) = deThe random variable E is described by the distribution (A · ]sin(e)|, fE(e) = f(x) = {" -TTProblem 3.5: The density function of a continuous random variable X is shown below. Find (a) the mode. (b) the median, and (c) the mean. [0.9391 sin(√2-1) for 0.8 < x < 2.2 elsewhereSuppose X and Y are two independent variables with variance 1. Let Z = X+bY where b > 0. If Cor(Z, Y ) = 1/2, what is the value of b?Suppose the random variable T is the length of life of an object (possibly the lifetime of an electrical component or of a subject given a particular treatment). The hazard function hr(t) associated with the random variable T is defined by hr(t) = lims-o- P(t ≤ TSSuppose that the probability that a patient admitted in a hospital is diagnosed with a certain type of cancer is 0.03. Suppose that on a given day io patients are admitted and X denotes the number of patients diagnosed with this type of cancer. The mean and the variance of X are: E(X)=0.4 and V(X)=0.384 O None of these O E(X)=0.5 and V(X)=0.475 O E(X)=0.3 and V(X)=0.291Problem 3.5: The density function of a continuous random variable X is shown below. Find (a) the mode. (b) the median, and (c) the mean. -{0.93⁹ (0.9391 sin(√2-1) for 0.8 < x < 2.2 f(x) = elsewhereQ. 6 Find 30th percentile, median and 70th percentile for the following probability density functions: X < 0 0s X < 1 0s X s 1.3 1.3 (i) fx(x) = 3x2 1s Xs 2 (ii) fx(x) = %3D %3D 2-x 1.3 < X s 2 14 otherwise 0.7 otherwiseSuppose that the probability that a patient admitted in a hospital is diagnosed with a certain type of cancer is 0.03. Suppose that on a given day 10 patients are admitted and X denotes the number of patients diagnosed with this type of cancer. The mean and the variance of X are: None of these E(X)=0.5 and V(X)=0.475 E(X)=0.4 and V(X)=0.384 E(X)=0.3 and V(X)=0.291SEE MORE QUESTIONSRecommended textbooks for youGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill