• ↑
  • Joint Probability Mass Function

Marginal Distributions

• We sum along one variable to get individual Probability Mass Functions for each component.
• $\begin{array}{ccccc}X=x& x_1& x_2& \dots & x_m\\ P(X=x)\end{array}$
• $\begin{array}{ccccc}Y=y& y_1& y_2& \dots & y_n\\ P(Y=y)\end{array}$
• Let $X=(X_2,X_2)$ be a Random Vector where:
  • $X_1=\{0,1,2\}$ this is number of coffees sold
  • $X_2=\{0,1\}$ this is the number of muffins sold
  • $X_1,X_2$ has the Joint Probability Mass Function
  • $\begin{array}{cccc}& & & x_1\\ & & 0& 1& 2\\ x_2& 0& 0.10& 0.15& 0.05& 0.3\\ & 1& 0.05& 0.30& 0.35& 0.7\\ & & 0.15& 0.45& 0.4& 1\end{array}$
  • Find the probability that someone buys $2$ coffees and $0$ muffins
  • $P(X_1=2,X_2=0)=0.05$
  • #tk exercise
  • Find the probability that $P(X_1\le 1,X_2\le 1)$
    • $P(X_1=\{0,1\},X_2=\{0,1\})=0.60$
  • Find the marginal distributions
    • Probability Mass Function of $X_1$
    • $\begin{array}{cccc}{X}_1=x_1& 0& 1& 2\\ P(X_1=x_1)& 0.15& 0.45& 0.4\end{array}$
      • These add up to one, so this is valid
    • $\begin{array}{ccc}{X}_2=x_2& 0& 1\\ P(X_2=x_2)& 0.3& 0.7\end{array}$
      • This adds up to one, so this is also valid
• Moving from Joint Distribution to a distributions of 1 variable
  • Bivariate to univariate
• Marginal Probability Functions
• Slide 20
• Definition 9
  • $f_1(x_1)=\int f(x_1,x_2)d{x}_2$
• Example:
  • $f(x_1,x_2)=3x_1$ for $0\le x_2\le x_1\le 1$
  • Find the marginal distributions
  • Marginal of $x_1$
    • $f_{x_1}(x_1)={\int }_{?}^{?}{f}_{x_1,x_2}(x_1,x_2)d{x}_2$
    • $f_{x_1}(x_1)={\int }_{?}^{?}3x_{1}d{x}_2$
    • $f_{x_1}(x_1)={\int }_{x_2=0}^{x_2=x_1}3x_{1}d{x}_2$
    • $f_{x_1}(x_1)=[3x_1{x}_2]{}_{x_2=0}^{x_2=x_1}$
    • $f_{x_1}(x_1)=3x_1^2$ for $0\le x_1\le 1$
  • Marginal of $x_2$
    • $f_{x_2}(x_2)={\int }_{?}^{?}{f}_{x_1,x_2}(x_1,x_2)d{x}_1$
    • $f_{x_2}(x_2)={\int }_{?}^{?}3x_{1}d{x}_1$
    • $f_{x_2}(x_2)={\int }_{x_1=x_2}^{x_1=1}3x_{1}d{x}_1$
    • $f_{x_2}(x_2)=\left[\frac{3}{2}{x}_1^2\right]{}_{x_1=x_2}^{x_1=1}$
    • $f_{x_2}(x_2)=\frac{3}{2}-\frac{3x_2^2}{2}$ for $0\le x_2\le 1$