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:

$$P(a\le X\le b)={\int }_a^{b}f(x)dx.$$

• • The total probability must sum to 1:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1.$$

Example: Find the Constant $A$

Given: $f(x)=A{x}^3,0\le x\le 1$.

• • The integral must equal 1:

$${\int }_0^{1}A{x}^{3}dx=1.$$

• • 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 $f(x)=4x^3,0\le x\le 1$.

• • 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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