I'm currently trying to catch up on Stochastics for university and I'm really stuck on this one although I feel like its not as difficult as I may think:
Let X, Y be random variables. The random variable X takes the values 1, 2 and 3 with probabilities
P(X = 1) = 0.5, P(X = 2) = 0.3 and P(X = 3) = 0.2 and Y takes the values 1 and 2 with
probabilities P(Y = 1) = 0.7 and P(Y = 2) = 0.3.
Moreover, it is known that P(X = 1, Y = 1) = 0.35 and P(X = 3, Y = 1) = 0.2.
a) Calculate the distribution functions of X and Y .
b) Compute the remaining probabilities P(X = i, Y = j) for i ∈ {1, 2, 3} and j ∈ {1, 2}.
Regarding a: I think I just need to see how its done once and than I'll probably get it but my University is not publishing the solutions or the script so my only options are googling and hoping to get it but I really dont :/
Regarding b: I get why P(X = 1, Y = 1) = 0.35 ( because 0.5*0.7 ) but why is P(X = 3, Y = 1) = 0.2 ( 0.2 * 0.7 ? )?
I know its alot and I'm not expecting full on solutions but maybe someone can explain the basic approach to solving the exercise?
I would greatly appreciate it 🙂
Best Answer
a) just calculate the CDF of X and Y thus refer to your textbook.
For example,
$$F_X(x)=0.5\cdot\mathbb{1}_{[1;2)}(x)+0.8\cdot\mathbb{1}_{[2;3)}(x)+\mathbb{1}_{[3;+\infty)}(x)$$
b)
These are your data expressed in a tabular mode
Can you complete the table?