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- If the joint probability distribution of X and Y is given by * f (x, y) x+y for x = 0,1,2,3; y = 0,1,2. Find the marginal distribution of X 30 f(x, y) 1 2 3 1 2 3 30 2 30 30 3 4 Y 1 30 30 30 30 3 4 2 30 30 30 30 1 2 3 1 3 f(x) 1 1 3 2 f(x) 1 7 2 10 10 3. 15 If none of the choices, fill the table 1 2 3 |1 2 3 f(x) 1 3 2 f(x) - 10 3 15Q.1 The probability mass function for a discrete random variable X is defined as ((1+0)" (^) 0x; x = 0, 1, 2, 3, ..., n fx(x) = {(1 + 0; e. w. where > 0. Show that it is probability mass function. Find its mean and variance.Let X; € {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid fx\p(x |p) = (1 – p)*-'p x-1 X1, X2, for 0If the joint probability distribution of X and Y is given by * x+y f(x,y) = for x = 30 0,1,2,3; y = 0,1,2. Find the marginal distribution of Y X f(x, y) 1 2 3 1 2 3 30 30 2 30 4 1 Y 30 2 30 3 30 4 30 2 30 30 30 30 Y 1 2 y 1 2 f(y) 1 1 3 f(y) 1 7 10 10 5 3 15 If none of the choices, fill the table Y 1 2 | 0 |1 |2 | 3 f(y) 1 3 f(x) - 10 3 152. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.7. If a random variable has the probability density function -1≤x≤3 otherwise f(x)=k(x²-1) =0Suppose the lifespan (in months) of a smartphone battery can be modeled as a continuous random variable with CDF F(x) = 1 − e-x/3 x ≥ 0 What is the probability that the battery lasts between 12 to 15 months?The pdf of random variable X is given as ƒx(x) = Find the i) Mean ii) Mean of the square [0.3507√x 0Let X; € {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2,..., Xn fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1,2, 3, ...} and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fp|X, (p | Xn).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