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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 2

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Consider the random variables $(X, Y)$ with the joint probability distribution given by:
$$
(X, Y)= \begin{cases}(1,4) & \text { with probability } \frac{1}{3} \\ (2,2) & \text { with probability } \frac{1}{3} \\ (4,1) & \text { with probability } \frac{1}{3}\end{cases}
$$
Determine the value of the cumulative distribution function $F_{X, Y}$ at the point $(3.5,4.5)$, i.e., $F_{X, Y}(3.5,4.5)$.

  1. $\frac{1}{3}$
  2. $\frac{2}{3}$
  3. $\frac{5}{6}$
  4. $\frac{1}{2}$
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Since $(3.5,4.5)$ is greater than all given points $(1,4),(2,2)$, and $(4,1)$, we include the probabilities of $(1,4)$ and $(2,2)$ :

$$
\begin{aligned}
F_{X, Y}(3.5,4.5) & =P((X, Y) \leq(3.5,4.5)) \\
& =P(1,4)+P(2,2) \\
& =\frac{1}{3}+\frac{1}{3}=\frac{2}{3}
\end{aligned}
$$
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