Consider continuous random variables X and Y that have the joint probability density function 2y f(x, y) = {3√e ²v if0 ≤ x ≤ 1 and 0 ≤ y < ∞ 0 otherwise. Compute the probability P(0 ≤ x ≤ 1,≤Y <∞). Please give your answer as a decimal number, rounded to four decimal places if necessary.
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- Find the moment-generating function of the contin-uous random variable X whose probability density is given by f(x) =1 for 0 < x < 10 elsewhere and use it to find μ1,μ2, and σ2.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}?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?
- 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 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 P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),Determine the conditional probability distribution of Y given that X = 1. Where the jointprobability density function is given by f(x,y)=1/64xy for 0 < x < 4 and 0 < y < 4.
- Let Y1 = 0.5, Y2 = 0.25, Y3 = 0.75, Y4 = 0.25 and Y5 = 1.25 be a random sample of width 5 selected from the population with the following probability density function. Which of the following is the estimation value obtained by the moment method for the unknown q parameter of this population?An insurance company finds that the proportion X of itssalespeople who sell more than $25,000 worth of insurance in a given week is a random variable with the beta probability density functionf(x) = 60x^3(1 - x)^2, 0 ≤ x ≤ 1.(a) Compute E(X ) and give an interpretation of thisquantity.(b) Compute Var(X ).Let the random variable Y denote the time (in minutes) for which a customer is waiting for the beginning of a service station since its arrival and let X denote the time (minutes) until the service is completed since its arrival at the service station. Since both X and Y measure the time since the arrival of the customer at the service station, always Y<X is true. The joint probability density function for X and Y is given as follows: fxy(x,y)=c(x+y) for 0<x<2 and 0<y<x what is the value of c? what is the covariance of X and Y? what is the correlation of X and Y?
- Suppose that X is a continuous random variable with density function f(x). If f(x)=k for −5≤x≤3 and f(x)=0 otherwise, determine the value of k.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?A continuous random variable has a density function f(x)= 2(5-x)/5 , where 2<x<3. Calculatethe following probability correct up to 3 decimal places, and make the graph for part (a) only in Answer sheet:P (x < 2.5)P (x > 2.2)P (2.1 ≤
