Discrete Random Variables Probability Exercise - How to approach this

Question

Below is the whole exercise that I need to solve.

Since this is from an online course and it's given with any other context, I need to figure what I need to learn in order to solve it.

Is it Binomial Distribution?

Any approach to the way to solve each of them would be awesome, because it would allow me to find a book and study the relevant part.

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Let $X$ and $Y$ be $2$ discrete random variables which probability set function is defined by:

• $f(1,3)=0.1$
• $f(1,5)=0.3$
• $f(2,3)=0.4$
• $f(2,5)=0.2$

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1. Without determining the marginal probability function for $X$ and $Y$, please solve, justifying your answers:

   • The value of $P(X=1,Y=<4)$ and $P(X=2|Y=3)$

   • The value of $E(X)$ and of $E(Y)$

   • The covariance between $X$ and $Y$

2. Find the marginal distribution function of $X$

Answer 1

Answering some of your questions:

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$P(A|B) = \begin{cases} \frac{P(A \cap B)}{P(B)} & \text{$P(B)>0$}\\ 0 & \text{$P(B)=0$}\\ \end{cases} $

Therefore:

• $P(X=2|Y=3) = \frac{P(X=2 \cap Y=3)}{P(Y=3)} = \frac{0.4}{0.1+0.4} = 0.8$

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$E(N) = \sum\limits_{i=1}^{k}N_iP_i = N_1P_1+N_2P_2+\dots+N_kP_k$

Therefore:

• $E(X) = 1\cdot0.1+1\cdot0.3+2\cdot0.4+2\cdot0.2 = 1.6$
• $E(Y) = 3\cdot0.1+5\cdot0.3+3\cdot0.4+5\cdot0.2 = 4.0$