Let \( X \) and \( Y \) be discrete random variables with joint probability den function \( f(x, y)=\left\{\begin{array}{ll}\frac{1}{21}(x+y) & x=1.2 .3 \\ 0 & y=1.2 \\ 0 & o \cdot w\end{array}\right. \) 1- The conditional density of X given Y 2- What is the conditional mean of X given \( \mathrm{Y}=\mathrm{y} \)

If you draw a card from a deck and do not replace it, what type of events does this represent when drawing a second card?

If you flip a coin and roll a die, what is the probability of getting heads and a 4?

Se a probabilidade de um evento A ocorrer é 0,3 e a probabilidade de um evento B ocorrer é 0,5, quais são as chances de ambos os eventos ocorrerem se eles forem independentes?

similarly \( P\left(A / E_{2}\right)=8 / 10 \), and \( P\left(A / E_{3}\right)=4 \) given that the ball drawn is red, what is the third bag by Baye's rule \( =\frac{1}{\frac{1}{3} \times \frac{3}{10}+\frac{1}{3} \times \frac{4}{10}+\frac{1}{3} \times \frac{4}{10}} \) Example 3.12 : Three machines \( E_{1}, E_{2}, E_{3} \) \( 25 \% \), respectively, of the total daily outpl the tubes produced one each of machine those produced on \( E_{3} \) are defective. If on

(Bayes' theorem) Example 3.11: Three bags contain 3 red, 7 black; 8 red, 2 black, and 4 re balls respectively. 1 of the bags is selected at random and a ball is draw the ball drawn is red, find the probability that it is drawn from the thirc Sol: Let \( E_{1}, E_{2}, E_{3} \), and A are the events defined as follows. \( E_{1}= \) First bag is chosen \( E_{2}= \) Second bag is chosen \( E_{3}= \) Third bag is chosen \( \mathrm{A}= \) Ball drawn is red Since there are three bags and one of the bags is chosen at random, so \( \mathrm{P}\left(E_{1}\right)=\mathrm{P}\left(E_{2}\right)=\mathrm{P}\left(E_{3}\right)=1 / 3 \) If \( E_{1} \) has already occurred, then first bag has been chosen which conta black balls. The probability of drawing 1 red ball from it is \( 3 / 10 \). So, \( \mathrm{P}( \) 37