I have two Binomial distributed random variables.
$$X_1\sim\text{Bi}\left(50,\theta_1\right)$$ $$X_2\sim\text{Bi}\left(50,\theta_2\right)$$
I sample from each distribution.
$$X_1=x_1=15$$ $$X_2=x_2=26$$
Question
formulate the joint likelihood function using the given information
Attempt 1
In this attempt I calculated the likelihood for each observation separately and multiplied them together. I am curious to know if I am on the right track?
First likelihood $$L\left(X_1=x_1|\theta_1,n\right)= {n\choose x_1}\theta_{1}^{x_1}(1-\theta_1)^{n-x_1} $$ $$n=50,x_1=15$$ $$L\left(X_1=15|\theta_1,n\right)= {50\choose 15}\theta_{1}^{15}(1-\theta_1)^{50-15} $$ $$L\left(X_1=15|\theta_1,n\right)= (2250829575120) \theta_{1}^{15}(1-\theta_1)^{35} $$
similarly, the second likelihood is:
$$L\left(X_2=26|\theta_2,n\right)= {50\choose 26}\theta_{2}^{26}(1-\theta_2)^{50-26} $$
Is the joint likelihood their product?
$$L_{\text{joint}}=L\left(X_1=15|\theta_1,n\right)\times L\left(X_2=26|\theta_2,n\right)$$