Let X1, X2, X3, ...., X₁ be a random sample from the distribution ƒ(x) = 9(1 − x)³ 0
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- 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 the random variables X and Y have joint probability density function f(x,y) given by: (image)Find: P(X < Y) = fX|Y=y (x)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)].
- 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?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?Suppose that X, Y are jointly continuous with joint probability density function f( x, y){ xe^-x(1+y), ifx >0 and y >00, otherwise. (a) Find the marginal density functions of X and Y. (b) Calculate the expectation E[XY]. (c) Calculate the expectation EIX/(1+ Y )1. (e) Determine if the random variables X and Y in this exercise are independent.
- The life (in years) of a laptop battery has a probability density function defined by P(x)=12e−x/2P(x)=12e-x/2for x in [0,∞)[0,∞). Find the probability that a randomly selected laptop battery will last between 3 and 8 years?Find a value of k that will make f a probability density function on the indicated interval.ƒ(x) = kx; [1, 5]Suppose that the joint density function of the random variables X and Y is f(x,y)=k(1+2y), if 1<x<13 and 0<y<1, and f(x,y)=0, otherwise. Show that the marginal distribution of X is g(x)=c, if 1<x<13, and g(x)=0 otherwise. Enter the value of c. Hint: Of course, first, you need to find the value of k. Round your answer to a number with two decimal digits after the decimal point. For example if your answer is 1/40, which is equal to 0.025, then you should enter 0.03. (Do NOT use decimal comma; 0,03 would be wrong.)
- Suppose that X is a continuous random variable with a probability density function is given by f(x)= 25 when x is between -2 and 2, and f(x)=0 otherwise. a.)Find E(X2), where X is raised to the power 2 b.) Find Var(2X+2)Let the continuous random variable X denote the current measured in a thin copper wire in milliamperes. Assume that the range of X is [4.9, 5.1] mA, and assume that the probability density function of X is f(x) = 5 for 4.9 <= x <= 5.1. What is the variance?Let X be a continuous random variable with density functionf(x) = 3x^-4, x ≥ 1. Compute E(X ) and Var(X ).