# Find $P(X=Y+1)$

Suppose that X and Y are integer valued random variables with joint probability mass function given by $$p_{X,Y}(a, b)=\begin{cases} \frac{1}{4a}, & 1\leq b\leq a\leq 4\\ 0, & \text{otherwise}. \end{cases}.$$

(b) Find the marginal probability mass functions of X and Y. (c) Find $$P(X=Y+1)$$

Since this is a discrete random variable, I want to construct a table of joint distribution to find the sum of rows, columns for pmf of X and Y, but I'm having trouble with it. Anyone can help with part (b) and (c)?

## 1 Answer

The table is not hard.   Each cell in 4 rows ($$1\leq a\leq 4$$) of $$a$$ columns ($$1\leq b\leq a$$), contains $$\tfrac 1{4a}$$, the rest of the cells are zero .

$$p_{X,Y}(a,b)=\tfrac 1{4a}\mathbf 1_{1\leq b\leq a\leq 4}\\\boxed{\begin{array}{c|c:c:c:c|c}a\backslash b & 1 & 2 & 3 & 4 &\\\hline 1 & \tfrac 14 &0&0&0\\ \hdashline 2 & \tfrac 18&\tfrac 18&0&0\\ \hdashline 3 &\tfrac 1{12}&\tfrac 1{12}&\tfrac 1{12}&0\\ \hdashline 4 & \tfrac 1{16}&\tfrac 1{16}&\tfrac 1{16}&\tfrac 1{16}\\\hline ~&&&&&1\end{array}}$$

• So $P(X=Y+1) = P(X=2, Y=1) + P(X=3, Y=2) + P(X=4, Y=3) = 1/8 + 1/12 +1/16 = 13/48$? – dxdydz Nov 14 '18 at 5:08
• Indeed. @dxdydz – Graham Kemp Nov 14 '18 at 11:59