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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f(x,y) gives the probability density of two continuous random variables X and Y occurring together
fX(x) or fY(y) gives the probability density of a single , 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 fX(x)=∫−∞∞f(x,y)dy
Properties of joint probability density functions:
Non-negative: f(x,y)≥0 for all x and y
Integrates to 1 over the entire domain: ∫−∞∞∫−∞∞f(x,y)dxdy=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)∈R)=∬Rf(x,y)dA
R is the region of interest in the xy-plane
dA represents the area element dxdy
Example: If the joint probability density function is f(x,y)=6xy over the region 0≤x≤1 and 0≤y≤1−x, the probability of (X,Y) falling within this triangular region is ∬R6xydA=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 of Y given X=x is denoted as fY∣X(y∣x)
Defined as fY∣X(y∣x)=fX(x)f(x,y), where f(x,y) is the joint probability density function and fX(x) is the marginal probability density function of X
Similarly, the conditional probability density function of X given Y=y is fX∣Y(x∣y)=fY(y)f(x,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≤Y≤d∣X=x)=∫cdfY∣X(y∣x)dy
Similarly, to calculate the probability of X falling within a range [a,b] given Y=y:
P(a≤X≤b∣Y=y)=∫abfX∣Y(x∣y)dx
Expected Value and Variance
Expected Value
(or ) of a X with probability density function f(x) is denoted as E(X) or μX
Calculated using the formula E(X)=∫−∞∞xf(x)dx
For a function g(X) of the random variable X, the expected value is given by E(g(X))=∫−∞∞g(x)f(x)dx
Expected value of a continuous random variable X with joint probability density function f(x,y) is E(X)=∫−∞∞∫−∞∞xf(x,y)dxdy
Variance
of a continuous random variable X measures the spread of the distribution around its expected value
Denoted as Var(X) or σX2
Calculated using the formula Var(X)=E((X−μX)2)=∫−∞∞(x−μX)2f(x)dx
Alternative formula for variance: Var(X)=E(X2)−(E(X))2
Where E(X2)=∫−∞∞x2f(x)dx is the expected value of X2
σX 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 σXY
Calculated using the formula Cov(X,Y)=E((X−μX)(Y−μY))=∫−∞∞∫−∞∞(x−μX)(y−μY)f(x,y)dxdy
Alternative formula for covariance: Cov(X,Y)=E(XY)−E(X)E(Y)
Where E(XY)=∫−∞∞∫−∞∞xyf(x,y)dxdy 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 ρXY
Calculated using the formula ρXY=σXσYCov(X,Y), where σX and σY are the standard deviations of X and Y, respectively
Properties of the correlation coefficient:
Always between -1 and 1, inclusive
ρXY=1 indicates a perfect positive linear relationship
ρXY=−1 indicates a perfect negative linear relationship