Continuous Random Variable and Probability Density Function

Data Science and A.I. Lecture Series

Continuous Random Variable and Probability Density Function

• • A random variable is continuous if it can take any real value within a given range.
  • Instead of probability mass function, we use probability density function (PDF), denoted by \( f(x) \).
  • The probability that \( X \) lies in an interval \( (a, b) \) is given by:

\[ P(a \leq X \leq b) = \int_{a}^{b} f(x)dx. \]

• • The total probability must sum to 1:

\[ \int_{-\infty}^{\infty} f(x)dx = 1. \]

Example: Find the Constant \( A \)

Given: \( f(x) = Ax^3, 0 \leq x \leq 1 \).

• • The integral must equal 1:

\[ \int_0^1 Ax^3 dx = 1. \]

• • Compute the integral:

\[ A \int_0^1 x^3 dx = A \left[ \frac{x^4}{4} \right]_0^1 = A \times \frac{1}{4}. \]

• • Solving for \( A \):

\[ A \times \frac{1}{4} = 1 \Rightarrow A = 4. \]

Example: Probability Computation

Find \( P(0.2 < X < 0.5) \) for \( f(x) = 4x^3, 0 \leq x \leq 1 \).

• • Compute the integral:

\[ P(0.2 < X < 0.5) = \int_{0.2}^{0.5} 4x^3 dx. \]

• • Evaluate:

\[ 4 \times \left[ \frac{x^4}{4} \right]_{0.2}^{0.5}. \]

• • Solve:

\[ \left[ x^4 \right]_{0.2}^{0.5} = (0.5)^4 – (0.2)^4 = 0.0625 – 0.0016 = 0.0609. \]

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Continuous Random Variable and Probability Density Function

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