Probability Density Function and CDF

Finding k

• The function f(x) is defined as follows: f(x) = kx, if 0 ≤ x ≤ 1; 0, otherwise
• For the function to be a probability density function, the integral of f(x) over all possible values of x must equal 1.
• Thus, the integral of f(x) from negative infinity to positive infinity must equal 1.
• ∫f(x) dx = ∫¹₀ kx dx = 1
• Evaluating the integral: k[x²/2]¹₀ = (k(1)² / 2 ) - (k(0)² / 2) = k/2 = 1
• Solving for k: k = 2

Finding F(x)

• F(x) represents the cumulative distribution function (CDF) related to the probability density function (PDF) denoted by f(x).
• F(x) is defined through integration: F(x) = ∫^x_-∞ f(t) dt.
• Since f(t) is defined in segments: F(x) = ∫^x₀ 2t dt when 0 ≤ x ≤ 1 and 0 otherwise
• Evaluating the integral ∫^x₀ 2t dt = t² |^x₀ = x²
• Thus, the resulting CDF function is: F(x) = x², for 0 ≤ x ≤ 1; 0, otherwise

This quiz explores the concepts of probability density functions (PDF) and cumulative distribution functions (CDF) through the function f(x) = kx. It includes topics such as integration of functions and finding constant values for PDFs. Test your understanding of these fundamental concepts in probability theory.