The time, in 100 hours, that a student uses her game console over a year is a random variable X with probability density function x if 0 < x < 1 f(x) = 2x if 1 < x < 2 0 otherwise. The power (in number of kilowatt hours) expended by the student's game console each year is 46X² + 24. For these problems, please ensure your answers are accurate to within 3 decimals. Part a) Find the mean amount of power expended by the student's game console per year. Part b) Find the variance of power expended by the student's game console per year. U

If X is a continuous variable in the range 3 > X > 0 and its distribution function is as follows: F ( x ) = k : ( x3 + x2) find the probability density function?

If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?

If the independent random variables X and Y havethe marginal densitiesf(x) =⎧⎪⎪⎨⎪⎪⎩12for 0 < x < 20 elsewhereπ(y) =⎧⎪⎪⎨⎪⎪⎩13for 0 < y < 30 elsewherefind(a) the joint probability density of X and Y;(b) the value of P(X2 + Y2 > 1).

Find μ,μ 2, and σ2 for the random variable X that has the probability density f(x) =⎧⎪⎨⎪⎩x2for 0 < x < 20 elsewhere

With the known values of a = 200,000  and b = 230,000,  the first equation for the probability density function for the sales price of a home is found as follows. f(x)         =            1/b – a,                                a ≤ x ≤ b = 1/230,000 − 200000,                                 200,000 ≤ x ≤ 230,000 = 1/30000,                                                    200,000 ≤ x ≤ 230,000   Everywhere else, the probability density function will just be 0. Therefore, the full probability density function for the sales price of a home follows.     f(x) =     _______________            200,000 ≤ x ≤ 230,000                                                                      elsewhere

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 the unknown X is ≥ 2  and has the probability density function fX(x)=Ce^−x,x ≥ 2.What is the numerical value of C?

suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)

Suppose that the probability density function of x is   fx=3x2, 0<x<1 0, elsewhere   Determine p(x < (1/3)), p((1/3) ≤ x < (2/3)), and p(x ≥ (2/3)) Determine the cumulative distribution function of x.