12.2 Probability and expected values

Probability and expected values are key concepts in calculus, bridging mathematics with real-world applications. They help us understand uncertainty and make predictions about random events, using double integrals to calculate probabilities over two-dimensional regions.

This topic dives into joint and marginal probability density functions, conditional probability, and expected values. We'll explore how to use double integrals to find probabilities, calculate conditional probabilities, and determine expected values and variances for continuous random variables.

Joint and Marginal Probability Density Functions

Defining Joint and Marginal Probability Density Functions

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• Joint probability density function $f(x,y)$ gives the probability density of two continuous random variables $X$ and $Y$ occurring together
• Marginal probability density function $f_X(x)$ or $f_Y(y)$ gives the probability density of a single random variable, either $X$ or $Y$, without considering the other variable
  • Can be obtained by integrating the joint probability density function over the range of the other variable
  • For example, the marginal probability density function of $X$ is given by $f_X(x) = \int_{-\infty}^{\infty} f(x,y) dy$
• Properties of joint probability density functions:
  • Non-negative: $f(x,y) \geq 0$ for all $x$ and $y$
  • Integrates to 1 over the entire domain: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) dx dy = 1$

Calculating Probabilities using Double Integrals

• Double integral in probability can be used to calculate the probability of two continuous random variables falling within a specific region
• Probability of $(X,Y)$ falling within a region $R$ is given by $P((X,Y) \in R) = \iint_R f(x,y) dA$
  • $R$ is the region of interest in the $xy$-plane
  • $dA$ represents the area element $dx dy$
• Example: If the joint probability density function is $f(x,y) = 6xy$ over the region $0 \leq x \leq 1$ and $0 \leq y \leq 1 - x$, the probability of $(X,Y)$ falling within this triangular region is $\iint_R 6xy dA = 1$

Conditional Probability

Definition and Formula

• Conditional probability measures the probability of an event occurring given that another event has already occurred
• For continuous random variables $X$ and $Y$, the conditional probability density function of $Y$ given $X=x$ is denoted as $f_{Y|X}(y|x)$
  • Defined as $f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)}$, where $f(x,y)$ is the joint probability density function and $f_X(x)$ is the marginal probability density function of $X$
• Similarly, the conditional probability density function of $X$ given $Y=y$ is $f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}$

Calculating Conditional Probabilities

• To calculate the probability of $Y$ falling within a range $[c,d]$ given $X=x$, integrate the conditional probability density function over that range:
  • $P(c \leq Y \leq d | X=x) = \int_c^d f_{Y|X}(y|x) dy$
• Similarly, to calculate the probability of $X$ falling within a range $[a,b]$ given $Y=y$:
  • $P(a \leq X \leq b | Y=y) = \int_a^b f_{X|Y}(x|y) dx$

Expected Value and Variance

Expected Value

• Expected value (or mean) of a continuous random variable $X$ with probability density function $f(x)$ is denoted as $E(X)$ or $\mu_X$
  • Calculated using the formula $E(X) = \int_{-\infty}^{\infty} x f(x) dx$
• For a function $g(X)$ of the random variable $X$, the expected value is given by $E(g(X)) = \int_{-\infty}^{\infty} g(x) f(x) dx$
• Expected value of a continuous random variable $X$ with joint probability density function $f(x,y)$ is $E(X) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x f(x,y) dx dy$

Variance

• Variance of a continuous random variable $X$ measures the spread of the distribution around its expected value
  • Denoted as $Var(X)$ or $\sigma_X^2$
  • Calculated using the formula $Var(X) = E((X - \mu_X)^2) = \int_{-\infty}^{\infty} (x - \mu_X)^2 f(x) dx$
• Alternative formula for variance: $Var(X) = E(X^2) - (E(X))^2$
  • Where $E(X^2) = \int_{-\infty}^{\infty} x^2 f(x) dx$ is the expected value of $X^2$
• Standard deviation $\sigma_X$ is the square root of the variance

Covariance and Correlation

Covariance

• Covariance measures the linear relationship between two continuous random variables $X$ and $Y$
  • Denoted as $Cov(X,Y)$ or $\sigma_{XY}$
  • Calculated using the formula $Cov(X,Y) = E((X - \mu_X)(Y - \mu_Y)) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x - \mu_X)(y - \mu_Y) f(x,y) dx dy$
• Alternative formula for covariance: $Cov(X,Y) = E(XY) - E(X)E(Y)$
  • Where $E(XY) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} xy f(x,y) dx dy$ is the expected value of the product $XY$
• Positive covariance indicates a positive linear relationship, negative covariance indicates a negative linear relationship, and zero covariance suggests no linear relationship

Correlation Coefficient

• Correlation coefficient measures the strength and direction of the linear relationship between two continuous random variables $X$ and $Y$
  • Denoted as $\rho_{XY}$
  • Calculated using the formula $\rho_{XY} = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}$, where $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$, respectively
• Properties of the correlation coefficient:
  • Always between -1 and 1, inclusive
  • $\rho_{XY} = 1$ indicates a perfect positive linear relationship
  • $\rho_{XY} = -1$ indicates a perfect negative linear relationship
  • $\rho_{XY} = 0$ suggests no linear relationship