Expectation of a Random Variable

Expectation of Random Variable $X$.

Expectation, also called expected value, is a fundamental concept in probability and statistics. It provides the average or mean value of a random variable if the experiment were repeated infinitely many time.

For Discrete Random Variable

$$\text{X is a discrete r.v, },E(X)=\sum x(PX=x)$$

For Continuous Random Variable

$$\text{X is a continuous r.v, },E(X)={\int }_{-\infty }^{\infty }x(PX=x)$$

Properties

• $E(aX+b)=aE(X)+b$ Where $a,b$ are constant.
• $E(a)=a$
• Let $Y=g(x)$
  • $E(Y)=E(g(x))=\sum g(x)\times P(x)$ $=\int g(x)\times P(x)$
    • $E(X^2)=\sum x^2\times P(x)$

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Variance of Random Variable $X$.

Formula for Variance

$$V(X)=E(X^2)-[E(X){]}^2$$

When $X$ is Discrete

$$V(X)=\sum [X^2\times P(X)]-[\sum X\times P(X){]}^2$$

When $X$ is Continuous

$$V(X)={\int }_{-\infty }^{\infty }{x}^{2}f(x)dx-{\left[{\int }_{-\infty }^{\infty }xf(x)dx\right]}^2$$

Alternate Formula for Variance

$$V(X)=E[(X-E(X){)}^2]$$

Properties of Variance

1. $V(a)=0$ where $a$ is a constant.
2. $V(aX)=a^2\cdot V(X)$ where $a$ is a scalar.

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Standard Deviation for Random Variable $X$.

Variance measures the spread or dispersion of a random variable’s values around its mean. It quantifies how much the values of a dataset or distribution deviate from the expected value (mean).

Formula for Standard Deviation

The standard deviation (${\sigma }_X$) is the square root of the variance:

$${\sigma }_X=\sqrt{V(X)}$$

When $X$ is Discrete

$${\sigma }_X=\sqrt{\sum [X^2\times P(X)]-{\left[\sum X\times P(X)\right]}^2}$$

When $X$ is Continuous

$${\sigma }_X=\sqrt{{\int }_{-\infty }^{\infty }{x}^{2}f(x)dx-{\left[{\int }_{-\infty }^{\infty }xf(x)dx\right]}^2}$$

Alternate Formula for Standard Deviation

$${\sigma }_X=\sqrt{E[(X-E(X){)}^2]}$$

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Standard $uv$ Rule Application in $E(X^2)$

Expectation Formula for Continuous Random Variable

The expectation of $X^2$ is given by:

$$E(X^2)=\int x^{2}f(x)dx$$

Applying the Standard $uv$ Rule

Using integration by parts ($uv$ rule), where $u=x^2$ and $v^{\prime }=f(x)$:

1. Let $u=x^2$, so $u^{\prime }=2x$.
2. Let $v^{\prime }=f(x)$, so $v=\int f(x)dx$.

Applying the integration by parts formula:

$$\int x^{2}f(x)dx=x^2\int f(x)dx-\int \left[2x\cdot \int f(x)dx\right]dx$$

Polynomial Behavior

Since $x^2$ is a polynomial, and differentiation reduces the degree of a polynomial, repeated application of the $uv$ rule will always result in a polynomial that eventually terminates. Therefore, the integration will conclude in a finite number of steps.

References

• Lectures at MPSTME
• Solve problems at Probability and Statistics Lecture 9

Information

• date: 2025.03.13
• time: 08:15