Independent solution

How to solve this Joint Discrete Distributions question

Setup

Setup

Use the admissible integer pairs only and normalize the joint mass function.

S={(x,y):0≤x≤y≤3}\mathcal S=\{(x,y):0\le x\le y\le3\}
1=c∑(x,y)∈S(x+2y)1=c\sum_{(x,y)\in\mathcal S}(x+2y)

Model

Model

The target count is the total minus the high-severity count, namely Y-X.

E[Y−X]=c∑(x,y)∈S(y−x)(x+2y)E[Y-X]=c\sum_{(x,y)\in\mathcal S}(y-x)(x+2y)

Compute

Compute

Evaluate the two finite weighted sums over the ten support points.

∑S(x+2y)=50⟹c=150\sum_{\mathcal S}(x+2y)=50\quad\Longrightarrow\quad c=\frac1{50}
∑S(y−x)(x+2y)=55\sum_{\mathcal S}(y-x)(x+2y)=55
E[Y−X]=5550=1.10E[Y-X]=\frac{55}{50}=1.10

Answer

Answer

The expected difference between the two counts is 1.10.

1.10(E)\boxed{1.10\quad\text{(E)}}