binomial distribution and function of several random variables

Let $$X_1$$ and $$X_2$$ be independent random variables with respective binomial distributions $$b(3,1/2)$$ and $$b(5,1/2)$$. Find $$P(X_1+X_2=7)$$.

I am thinking of an example I pulled from my notes: Ex: let $$X_1$$ ~ Poisson $$(\lambda_1=2)$$ and $$X_2$$~Poisson$$(\lambda_2=3)$$ be independent, then $$P(X_1+X_2=1)=P(X_1=0,X_2=1)+P(X_1=1,X_2=0)=(\lambda_1+\lambda_2)e^{-(\lambda_1+\lambda_2)}$$

So my thoughts for this question: $$P(x,p)=\binom{n}{x}(p)^x (1-p)^{n-x}$$. So for this question, should I cconsider by cases, like $$X_1=1, X_2=3$$ etc, and since they are independent, then I would add those cases together?

• Think what is the distribution of $X_1+X_2$, you need not formulas for it. – kludg Mar 10 '20 at 3:02
• can you specify? @kludg – shine Mar 10 '20 at 3:08

The binomial distribution with amount $$n$$ and success rate $$p$$, is more often denoted $$\mathcal{Bin}(n,p)$$.

$$Z\sim\mathcal{Bin}(n,p)$$ denotes that random variable, $$Z$$ is a count of successes among $$n$$ independent Bernoulli trials with identical success rate $$p$$.

So given that the independent random variables are $$X_1\sim\mathcal{Bin}(3, 1/2)$$ and $$X_2\sim\mathcal{Bin}(5,1/2)$$, what does $$X_1+X_2$$ count?   What is the distribution for this?

$$X_1$$ counts successes among $$3$$ independent Bernoulli trials with identical success rate $$1/2$$, and independently $$X_2$$ counts successes among $$7$$ independent Bernoulli trials with identical success rate $$1/2$$.

So $$X_1+X_2$$ counts...

Use this.

Okay, it is possible to use the Law of Total Probability.

Now, when the sum is $$7$$ the minimum $$X_1$$ may be is $$2$$ (because the maximum $$X_2$$ may be is $$5$$), so …

\begin{align}\mathsf P(X_1+X_2=7)~&=~\sum_{k=2}^3\mathsf P(X_1=k)~\mathsf P(X_2=7-k)\\[1ex]&=~\mathsf P(X_1=2)\,\mathsf P(X_2=5)+\mathsf P(X_1=3)\,\mathsf P(X_2=4)\\[1ex]&~~\vdots\end{align}

• I am still not sure of this , can you specify even more on counting $X_1+X_2$ and its distribution? – shine Mar 10 '20 at 3:23
• @shine … No. .. – Graham Kemp Mar 10 '20 at 3:47