5.3 Expected value and variance of continuous random variables

Continuous random variables are key in probability theory. They help us model real-world phenomena that can take on any value within a range. Understanding their expected value and variance is crucial for analyzing and predicting outcomes in various fields.

Expected value gives us the average outcome, while variance measures spread. These concepts apply to many distributions like uniform, exponential, and normal. Knowing how to calculate and interpret these values is essential for making informed decisions based on probabilistic models.

Expected Value and Variance of Continuous Random Variables

Expected value of continuous variables

Top images from around the web for Expected value of continuous variables

• 

  Probability density function - Wikipedia View original

  Is this image relevant?

• 

  probability - The random variable X has density function given by... Calculate the expectation ... View original

  Is this image relevant?

• 

  integration - Expectation of a Standard Normal Random Variable - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Probability density function - Wikipedia View original

  Is this image relevant?

• 

  probability - The random variable X has density function given by... Calculate the expectation ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Expected value of continuous variables

• 

  Probability density function - Wikipedia View original

  Is this image relevant?

• 

  probability - The random variable X has density function given by... Calculate the expectation ... View original

  Is this image relevant?

• 

  integration - Expectation of a Standard Normal Random Variable - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Probability density function - Wikipedia View original

  Is this image relevant?

• 

  probability - The random variable X has density function given by... Calculate the expectation ... View original

  Is this image relevant?

1 of 3

• Expected value (mean) of a continuous random variable $X$ with probability density function $f(x)$ calculated by integrating the product of $x$ and $f(x)$ over the entire range of $X$ ($-\infty$ to $\infty$)
  • Formula: $E[X] = \int_{-\infty}^{\infty} x f(x) dx$
  • Represents the weighted average value of the random variable considering its probability distribution
• Linearity of expectation holds for continuous random variables
  • For continuous random variables $X$ and $Y$ and constants $a$ and $b$: $E[aX + bY] = aE[X] + bE[Y]$
  • Allows breaking down the expected value of a sum into the sum of expected values
• Expected value of a function of a continuous random variable $g(X)$ obtained by integrating the product of $g(x)$ and $f(x)$ over the entire range of $X$
  • Formula: $E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) dx$
  • Useful for calculating moments and other quantities related to the distribution

Variance and standard deviation calculation

• Variance of a continuous random variable $X$ with probability density function $f(x)$ measures the average squared deviation from the mean
  • Formula: $Var(X) = E[(X - E[X])^2] = \int_{-\infty}^{\infty} (x - E[X])^2 f(x) dx$
  • Quantifies the spread or dispersion of the random variable around its mean
• Standard deviation is the square root of the variance
  • Formula: $\sigma_X = \sqrt{Var(X)}$
  • Provides a more interpretable measure of dispersion in the same units as the random variable
• Alternative formula for variance using the second moment $E[X^2]$ and the squared mean $E[X]$
  • Formula: $Var(X) = E[X^2] - (E[X])^2$
  • Simplifies calculations when the second moment is easier to compute than the integral with $(x - E[X])^2$

Properties of continuous random variables

• Linearity of variance for independent continuous random variables $X$ and $Y$ and constants $a$ and $b$
  • $Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)$
  • Allows decomposing the variance of a sum of independent variables
• Scaling property of variance for a constant $a$
  • $Var(aX) = a^2 Var(X)$
  • Variance scales quadratically with the constant factor
• Shifting property of variance for a constant $b$
  • $Var(X + b) = Var(X)$
  • Adding a constant to a random variable does not change its variance

Applications of continuous probability distributions

• Uniform distribution $U(a, b)$ has a constant probability density between $a$ and $b$
  • $E[X] = \frac{a + b}{2}$, the midpoint of the interval
  • $Var(X) = \frac{(b - a)^2}{12}$, proportional to the squared length of the interval
• Exponential distribution $Exp(\lambda)$ models waiting times and inter-arrival times
  • $E[X] = \frac{1}{\lambda}$, the reciprocal of the rate parameter
  • $Var(X) = \frac{1}{\lambda^2}$, inversely proportional to the squared rate
• Normal distribution $N(\mu, \sigma^2)$ is a common bell-shaped distribution
  • $E[X] = \mu$, the mean parameter
  • $Var(X) = \sigma^2$, the variance parameter
• Gamma distribution $Gamma(\alpha, \beta)$ generalizes the exponential and models waiting times for $\alpha$ events
  • $E[X] = \alpha \beta$, the product of the shape and scale parameters
  • $Var(X) = \alpha \beta^2$, proportional to the shape and squared scale