Math Problem Statement

Given a probability density function f(x) = K * e^(-(x-3)^2 / 2) for 0 <= x <= 6, determine the value of K and calculate the mean E(X).

Solution

The text extraction from the image didn't work well. Let me instead describe and interpret the image content directly.

From what I can see in the image, it appears that we are dealing with a problem involving a probability density function (PDF), particularly in the context of statistics. The problem asks for two things:

  1. Determine the value of KK (a constant).
  2. Calculate the mean E(X)E(X).

The provided probability density function (PDF) is: f(x)=K⋅e−(x−3)22, for 0≤x≤6.f(x) = K \cdot e^{-\frac{(x - 3)^2}{2}}, \text{ for } 0 \leq x \leq 6.

This appears to be a Gaussian-like (normal distribution) PDF, with KK being a normalizing constant. I will walk through the steps necessary to determine KK and calculate E(X)E(X) based on this interpretation.

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Math Problem Analysis

Mathematical Concepts

Probability Density Functions
Normalization of Functions
Gaussian Distribution
Mean of Continuous Random Variables

Formulas

f(x) = K * e^(-(x-3)^2 / 2)
Mean E(X) = ∫ x * f(x) dx
Normalization Condition: ∫ f(x) dx = 1

Theorems

Normalization of Probability Density Functions
Gaussian Distribution

Suitable Grade Level

Undergraduate

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