Suppose that the joint probability density function of X and Y is fxy(x,y) = 78.125(x² - y²) e-5×, for 0
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- If X has probability density function f(x) = X/32 on [0, 8], find the expected value and the standard deviation of X. Give your answer either as a irreducible fraction or a decimal number accurate to 2 decimal places. E(X) = std(X) =For what value of c is the function f(x) = {c/x³ if x ≥ 1 {gx3 otherwise a probability density function?The cumulative distribution function of the continuous uniform distribution between con- stants a and b is given by F(x) = P(X ≤ x) = x-a - a x b (a) The probability density function is f(x) = F(r). Find the form of f(x). (b) Find the derivative of f(x) for x = [a, b]. Is f(x) decreasing, increasing or flat in this region? (c) Does f(x) have a single maximum in the region [a,b]? If so, what is it, or if not, why not?
- For the joint probability density function f(x.y)= k(x² +y²), -24 Consider the following two functions, each having an unknown parameter c. c (x² + 4x + 7) -5 < x < 5 f (x) = otherwise, 100 (x – c)² – 1) -5The exponential distribution is a probability distribution for non-negative real numbers. It is often used to model waiting or survival times. The version that we will look at has a probability density function of the form p(y|v) = exp(-e¯ºy — v) (1) where y R+, i.e., y can take on the values of non-negative real numbers. In this form it has one parameter: a log-scale parameter v. If a random variable follows an exponential distribution with log-scale v we say that Y~ Exp(v). If Y~ Exp(v), then E [Y] =e" and V [Y] = e²v. 1. Produce a plot of the exponential probability density function (1) for the values y E (0, 10), for v = 1, v = 0.5 and v= 2. Ensure the graph is readable, the axis are labeled appropriately and a legend is included. ta il 2. Imagine we are given a sample of n observations y = (y₁,..., yn). Write down the joint proba- bility of this sample of data, under the assumption that it came from an exponential distribution with log-scale parameter v (i.e., write down the…4. Let F(x) = S f(z)dz where f(x) is a valid probability density function. Find an expression for d F'(x) = F(x) in terms of the function f. Hint: Look up the Fundamental Theorem of Calculus on Wikipedia and read the corollary.Suppose X has probability density function fx (x) =e for x > 0 and fx (x) = 0 otherwise . Р(2 < X<7) —?The continuous random variable X has the following probability density function (pdf), for some positive constant c, for 0 < x < c. 3 (1+x)³ f(x)= (a). Prove that c= √3-1. (a) Accept c = √√3-1. Plot f(x) in the range. (b) Find E(X); Var (X) using numerical integration. (c) Use Inversion method, find 20 random numbers from X =The time, X, for clearing side effects of water purification, in days, has the following cumulative distribution F: 1 F(x) = 0 for x 9 2 a) What is the probability density function for X for 1 5? c) What is the probability X 7? e) What is the probability that X > 6.3? f) What is the probability that X > 7 given the X > 6.3? g) Calculate the 70th percentile of X. h) What is the expected value of X? i) What is the expected value of x2 ? j) What is the variance of X? k) What is the probability that X is more than 0.1 below its expected value?Write all steps[1] The probability density function of a random variable X is SC(x + 4), -4 4.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