Suppose X₁ is a random variable which takes values (x₁,x) and X₂ is a random variable which takes values (x,x). Below is the table with values for joint probability P(X1, X2): x2 0.3 0.1 x2 0.4 0.2 which of the following statements is correct? A: P(X₁ = x₁, X₂ = x₂) = 0.3, B: P(X₁ = x₁) = 0.4, C: P(X₂ = x₂) = 0.2. Consider a coin experiment, where out of 10 trials, 3 successes were observed. Suppose that the Beta-Binomial model (with a uniform prior) was selected to fit the data. Which of the below is TRUE? Hint: Recall that the Probability Mass Function (PMF) for the Binomial distribution is P(x = k) n! (n- -k)!k!! Beta distribution is F-1 (1-0)³–¹1) T(a+B) T(a)r (3) = 0k (1 – 0)n—k, and the Probability Density Function (PDF) for the A: posterior p(x) ∞ Beta(4,8) B: posterior p(x) x Beta(12,5)

Peter fails quizzes with probability 1/4, independent of other quizzes. What is the probability that the second and third time Dave fails a quiz will occur when he takes his eight and ninth quizzes respectively?

Suppose that each time Bar-bie throws a dart, she has a 3/4 probability of getting a bullseye. If Barbie shoots three darts, what is the probability that she gets a bullseye on at least two of them?

Suppose that X is a N(3,4) random variable and suppose that Y=5X+2. Determine P(Y>18.5).

Do question 2

A congressional committee is composed of 10 members: 6 Republicans and 4 Democrats. On any given vote, members vote independently. Whereas Republicans always vote their party line, Democrats vote their party line with probability .8, and vote Republican with probability .2. On a rainy day only 3 committee members (selected w/o replacement) were present. Let                      X = number of Democrats present                            Y = Number who voted Democratic that day   Display the table of joint probabilities fX,Y(i,j)  for     0 ≤ j ≤ i, for i=0,1,2,3. We can say that p(i,j)=P[x=i, Y=j].