Let X and Y be two independent random variables with densities fX(x) = e^(-x), for x>0 and fY(y) = e^y, for y<0, respectively. Determine the density of X+

Let x and y have joint density function * + 2y), for 0 3 Enter the exact answer. (1>}) - 3 1 (b) Find the probability thatx < + y. Enter the exact answer. px +

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?

The bearing capacity of the soil under a foundation is measured to range from 200 kPA to 450 kPA. The probability density within this range is given by engineering office as f(x) = 3 ,200< x < 450 450 ,otherwise If you know that the column on which stands is designed to carry a load 275 kPA, what is the probability of the failure of the foundation? Do you think redesigning project is necessary?

Let X be a uniform random variable on the interval (-3,3) and let Y = X^2 A) find E(Y). B) find the probability density function of Y

A continuous variable X has a probability density function given by p(x)=Ax^2 for 0≤x≤4, and p(x)=0 otherwise. What is the value of the normalisation constant, A?