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- Show that f(x)f(x) is a probability density function, Find E(X)E(X) and Var(X)Var(X).Let f(x) = k(4x –x²) if 03. (a) For what value of k is f(x) a probability density function? (b) For that value of k, find P (x>2). (c) Find the mean.The random variables X and Y have a joint probability density function given by f(x, y) = way, 0 < x < 3 and 1 < y < x, and 0 otherwise.
- Show that the following are the probability density functions: fi(x) = e-*I(0,c0) (x) f2(x) = 2e¬*I(o,00) f(x) = (0 + 1)f1 (x) – Of2(x) 0 < 0 < 1Let f(x) = k(3x - x²) if 0 ≤ x ≤ 3 and f(x) = 0 if x 3. (a) For what value of k is f a probability density function? k= (b) For that value of k, find P(X > 1). P(X > 1) = (c) Find the mean. H =Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-∞,0) is a function f such that f(2) 20 and Lsla) = 1. (a) Determine which of the following functions are probability density functions on the (-00, 00). (1-1 00 (b) We can also use probability density functions to find the expected value of the outcomes of the event - if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. zf(z) dz yields the expected value for a density f(x) with domain on the real mumbers.) Find the expected value for one of the valid probability densities above.
- The probability density function of a distribution is given by x f(x) = exp(-7). Use differentiation to show that the probability density function has a maximum at x = 0. The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1], q = 1 - p and n is a positive integer. Use the moment generating function to find the mean and variance of the distribution in terms of p, q and n.Consider the probability density function 0 if x 1.Consider the probability density function 0 if x 1.
- Let f(x) = k(3x − x2) if 0 ≤ x ≤ 3 and f(x) = 0 if x < 0 or x > 3. (a) For what value of k is f a probability density function? k = (b) For that value of k, find P(X > 1). P(X > 1) = (c) Find the mean. ? =Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-∞,00) is a function f such that f(x) > 0 and | f(x) = 1. (a) Determine which of the following functions are probability density functions on the (-00, 0). -1 0 0 (b) We can also use probability density functions to find the expected value of the outcomes of the event - if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. , xf (x) dx yields the expected value for a density f(x) with domain on the real numbers.) Find the expected value for one of the valid probability densities above.For the probability density function f(x) = 3x^2 on [0,1], find: V(X)