Question

F(x)=1/2x+3. Найти E(f). Помогите.

Answer 1

От минус бесконечности до плюс бесконечности, если кратко, то упринадлежит R

Answer 2

To find the expected value E(f) of the function f(x) = (1/2)x + 3, we need to take the average value of f(x) over its entire domain. The expected value is also known as the mean or the average value of a function in the context of probability.

The formula for the expected value E(f) of a continuous function f(x) over an interval [a, b] is given by the definite integral of f(x) over that interval, divided by the width of the interval (b - a):

E(f) = (1 / (b - a)) * ∫[a to b] f(x) dx

In the case of the function f(x) = (1/2)x + 3, there's no specified interval, so we'll assume that we want to find the expected value over the entire real line.

Since the function is continuous and defined for all real numbers, we'll take the integral over the entire real line:

E(f) = (1 / (b - a)) * ∫[-∞ to +∞] ((1/2)x + 3) dx

Now, let's calculate the integral:

E(f) = (1 / (b - a)) * ∫[-∞ to +∞] ((1/2)x + 3) dx = (1 / (b - a)) * (∫[-∞ to +∞] (1/2)x dx + ∫[-∞ to +∞] 3 dx)

To compute each integral, we'll use the basic rules of integration:

∫(1/2)x dx = (1/2) * (x^2 / 2) + C1 = (1/4)x^2 + C1 ∫3 dx = 3x + C2

Now, let's continue with the calculations:

E(f) = (1 / (b - a)) * (∫[-∞ to +∞] (1/2)x dx + ∫[-∞ to +∞] 3 dx) = (1 / (b - a)) * ((1/4)x^2 + C1 + 3x + C2)

Since we're considering the entire real line, the interval's width (b - a) will be infinity. Hence, we can ignore it for this calculation.

E(f) = (1/4)x^2 + 3x + C

Where C is the constant of integration. Since the expected value is a constant, the value of C doesn't matter in this context.

So, the expected value of f(x) = (1/2)x + 3 is given by E(f) = (1/4)x^2 + 3x + C.