关于Bernoulli(θ/3)的含义及θ/3推导逻辑的技术疑问

Hey there! Let's unpack this clearly, step by step, so it clicks.

First, let's break down what Bernoulli(θ/3) means:

• A Bernoulli distribution is the standard model for any trial with exactly two outcomes—think "success" vs "failure". Here, "success" is defined as drawing a blue ball (so Xi=1), and "failure" is drawing a red ball (so Xi=0).
• The value inside the parentheses (θ/3 here) is the probability of success on a single trial. So Bernoulli(θ/3) tells us that every time you draw a ball, the chance it's blue is θ/3, and the chance it's red is 1 - θ/3.

Now, why does the success probability equal θ/3? Let's tie it straight to your problem setup:

• θ represents the total number of blue balls in the bag, and the bag has exactly 3 balls total.
• When you draw a ball at random (with replacement, so the bag's composition never changes between draws), the basic probability of picking a blue ball is just the count of blue balls divided by the total number of balls. That's fundamental probability logic!
• So that's θ (blue balls) / 3 (total balls) = θ/3. Since each draw is independent (you put the ball back after each pick), every Xi follows the same Bernoulli(θ/3) distribution.

To make it super concrete with examples:

• If θ=0 (no blue balls), θ/3=0—you'll never draw a blue ball, so every Xi=0.
• If θ=3 (all blue balls), θ/3=1—you'll always draw a blue ball, so every Xi=1.
• If θ=1 (one blue, two red), θ/3=1/3—you have a 1 in 3 chance of drawing blue each time.

备注：内容来源于stack exchange，提问作者ryan chandra