Question

2. Consider $Y_1,...,Y_n \sim Bern(\theta), 0 \le \theta \le 1$. That is each $Y_i$ is a Bernoulli random variable $$Y_i = \begin{cases} 1, & \text{with probability } \theta, \\ 0, & \text{with probability } 1-\theta. \end{cases}$$ for $i = 1....,n$. Suppose one observed $Y_1 = y_1,...,Y_n = y_n$. Derive the posterior distribution of $\theta|Y_1 = y_1,...,Y_N = y_n$ using a prior distribution $\theta \sim beta(a, b)$.

Name: consider y1dotsynberntheta 0theta 1 that is each yi is a bernoulli random variable yi1 with probability theta 0 with probability 1 theta for i1dotsn suppose one observed y1y1dotsynyn 84508
Uploaded: 2023-12-20T13:44:49-08:00
Duration: 2 min 30 s
Channel: Adi S
Description: consider y1dotsynberntheta 0theta 1 that is each yi is a bernoulli random variable yi1 with probability theta 0 with probability 1 theta for i1dotsn suppose one observed y1y1dotsynyn 84508

          2. Consider $Y_1,...,Y_n \sim Bern(\theta), 0 \le \theta \le 1$. That is each $Y_i$ is a Bernoulli
random variable
$$Y_i = \begin{cases}
1, & \text{with probability } \theta, \\
0, & \text{with probability } 1-\theta.
\end{cases}$$
for $i = 1....,n$.
Suppose one observed $Y_1 = y_1,...,Y_n = y_n$. Derive the posterior distribution
of $\theta|Y_1 = y_1,...,Y_N = y_n$ using a prior distribution $\theta \sim beta(a, b)$.

consider y1dotsynberntheta 0theta 1 that is each yi is a bernoulli random variable yi1 with probability theta 0 with probability 1 theta for i1dotsn suppose one observed y1y1dotsynyn 84508

Added by Sara L.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

12/20/2023

Step 1

..,y_n) = \prod_{i=1}^n \theta^{y_i}(1-\theta)^{1-y_i} = \theta^{\sum_{i=1}^n y_i}(1-\theta)^{n-\sum_{i=1}^n y_i}$$ Show more…

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2. Consider $Y_1,...,Y_n \sim Bern(\theta), 0 \le \theta \le 1$. That is each $Y_i$ is a Bernoulli random variable $$Y_i = \begin{cases} 1, & \text{with probability } \theta, \\ 0, & \text{with probability } 1-\theta. \end{cases}$$ for $i = 1....,n$. Suppose one observed $Y_1 = y_1,...,Y_n = y_n$. Derive the posterior distribution of $\theta|Y_1 = y_1,...,Y_N = y_n$ using a prior distribution $\theta \sim beta(a, b)$.