-0.6z A continuous random variable X has cumulative distribution function F(x) = 1 - e a) Find the density function of X. f(x) = b) Find the probability that X < 1. c) Find the probability that X > 2. d) Find the probability that X = 5. IO T 0.
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- Suppose that two-dimensional continuous random variable (X, Y) has joint probability density function given by f(x,y) = 24xy, x is less than equal to 1 and greater than equal to 0, y is less than equal to 1 and greater than equal to 0, x+y is less than equal to 1 and greater than equal to 0. Check that E(Y) = E[E(Y|X)] and V(Y) = E[V(Y|X)] + V[E(Y|X)].Suppose a continuous random variable X~Fx(x): f(x,y) = {1/4e^-1x/4, if x≥0 0, x<0} What is the cumulative density function of Y=min{2,X}?Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise Find the constant c, Find the marginal distribution functions(cdfs) for X and Y, Find the marginal density functions(pdfs) for X and Y, P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),
- 1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.X is a random variable whose cumulative distribution function F(x) satisfies F(1) = 13, F(2) = 34, and F(3) = 1 • Make X discrete and give its probability mass function.• Make X continuous and give its probability density function.Let X and Y be independent, uniformly distributed random variables in the range (0 1). If Z = X + Y, what is the probability P[Z < 0.5]? NOTE: The probability density function of a regular random variable defined in the range (0,1) is constant and takes the value 1.
- A continuous random variable X has a uniform distribution on the interval [−3,3]. Sketch the graph of its density function.6.) Suppose X is continuously uniformly distributed on [−2, 2]. Let Y = X2. What is the density function of Y? What is the expected value of Y?On a production line, parts are produced with a certain average size, but the exact size of each part varies due to the imprecision of the production process. Suppose that the difference between the size of the pieces produced (in millimeters) and the average size, which we will call production error, can be modeled as a continuous random variable X with a probability density function given by f(x) = 2, 5e^(-5|x|), for x E R (is in the image). Parts where the production error is less than -0.46 mm or greater than 0.46 mm should be discarded. Calculate (approximating to 4 decimal places): a) What is the proportion of parts that the company discards in its production process? b) What is the proportion of parts produced where the production error is positive? c) Knowing that for a given part the production error is positive, what is the probability of this part being discarded?
- Let X and Y be random variables with the joint density function f(x,y)=x+y, if x,y element of [0,1], and f(x,y)=0,elsewhere. Find the expected value of the random variable Z = 10X+14Y.Assume that X and Y are jointly continuous random variables with joint probabilitydensity function given by A. Find the marginal density functions for X and Y . B. Find E[X] and E[Y ]. C. Find Cov(X, Y ).Consider two random variables X and Y whose joint probability density function is given byf_X,Y (x, y) = c if x + y ≥ 1, x ≤ 1, and y ≤ 1,0 otherwise What is the value of c? State the limits of integration for both x and y. f_X,Y (x, y) = c if x + y <=2 1, x ≤ 2, and y ≤ 2,0 otherwise What is the value of c? State the limits of integration for both x and y.