Question

Найти E(y) y=2cosx+5

Answer 1

Область определения функции

Answer 2

To find the expected value E(y) of the random variable y = 2cos(x) + 5, we need to integrate y with respect to its probability density function (PDF) over the entire domain of x.

Since you haven't provided any specific information about the distribution of x, I'll assume that x is a continuous random variable with a uniform distribution over the interval [a, b].

Let's calculate E(y) step by step:

Step 1: Find the PDF of x. For a continuous uniform distribution over [a, b], the PDF is given by:

f(x) = 1 / (b - a) if a ≤ x ≤ b 0 otherwise

Step 2: Find the expected value E(y). E(y) = ∫[a to b] (y * f(x)) dx

Since y = 2cos(x) + 5 and f(x) = 1 / (b - a), we have:

E(y) = ∫[a to b] [(2cos(x) + 5) / (b - a)] dx

Step 3: Calculate the integral.

E(y) = (1 / (b - a)) * ∫[a to b] (2cos(x) + 5) dx = (1 / (b - a)) * [2∫[a to b] cos(x) dx + 5∫[a to b] dx]

The integral of cos(x) is sin(x), and the integral of dx is x:

E(y) = (1 / (b - a)) * [2[sin(b) - sin(a)] + 5[b - a]]

Step 4: Simplify the expression.

E(y) = (2 / (b - a)) * [sin(b) - sin(a)] + 5

This is the expected value E(y) for the given random variable y = 2cos(x) + 5, assuming x follows a continuous uniform distribution over the interval [a, b]. If there's a different distribution for x or if you have specific values for a and b, please provide that information for a more precise calculation.