Q.4.24 - A First Course in Probability

Question

Ten balls are to be distributed among 5 urns, with each ball going into urn i with probability pi ,$\sum _{i=1}^5{p}_i=1$

Let Xi denote the number of balls that go into urn i. Assume that events corresponding to the locations of different balls are independent.

1. What type of random variable is Xi? Be as specific as possible. 
2.  for $i\ne j$,what type of random variable is$X_i+X_j$
3.  find $P\left\{X_1+X_2+X_3=7\right\}$

Step-by-Step Solution

Answer

In the given information the answer of part

1.  $X_i$ is $X_i~\mathrm{Binom}\left(10,p_i\right)$
2. $X_i+X_j$ is $X_i+X_j~\mathrm{Binom}\left(10,p_i+p_j\right)$
3. The value of $P\left\{X_1+X_2+X_3=7\right\}$ is $\left(\begin{array}{c}10\\ 7\end{array}\right){\left(p_1+p_2+p_3\right)}^7{\left(p_4+p_5\right)}^3$

Step 1:Given Information (Part-a)

Given in the question that,$X_i$ indicate the number of balls that go into urn $i$ .Ten balls are to be distributed among 5 urns, with each ball going into urn $i$ with probability pi ,$\sum _{i=1}^3{p}_i=1$

We have to find what type of random variable is $X_i$ 

Step 2:Explanation (Part-a)

Observe that each of the ball goes to urn $i$ with the probability $p_i$ and does not go with probability $1-p_i$. Because of the fact that each ball chooses its urn independently from every other, we have that  $X_i~Binom\left(10,p_i\right)$

Step 3:Final Answer (Part-a)

$X_i$ is the $X_i~Binom\left(10,p_i\right)$

Step 4:Given Information (Part-b)

Given in the question that $X_i$ denote the number of balls that go into urn $i$ .Ten balls are to be distributed among 5 urns, with each ball going into urn $i$ with probability pi  ,$\sum _{i=1}^5{p}_i=1$

We need to find what type of random variable is $X_i+X_j$

Step 5:Explanation (Part-b)

Observe that random variable $X_i+X_j$ marks number of balls that goes to the urn $i$ or to the urn $j$. Since every ball goes to one and only one urn, the probability that some ball goes to $i^{th}$  or $j^{th}$ urn is  $p_i+p_j$.Because of the independence, we have that$X_i+X_j~Binom\left(10,p_i+p_j\right)$

Step 6 :Final Answer(Part-b)

$x_i+x_j$ is  $X_i+X_j~Binom\left(10,p_i+p_j\right)$

Step 7:Given information (part-c)

Given in the question that, $X_i$ denote the number of balls that go into urn $i$ .Ten balls are to be distributed among 5 urns, with each ball going into urn $i$ with probability pi ,$\sum _{i=1}^3{p}_i=1$

We need to find $P\left\{X_1+X_2+X_3=7\right\}$

Step 8: Explanation(Part c)

Using the similar argument as in (a) and (b) ,we have that$X_1+X_2+X_3~Binom\left(10,p_1+p_2+p_3\right)$

Hence, we have that

$P\left(X_1+X_2+X_3=7\right)=\left(\begin{array}{c}10\\ 7\end{array}\right){\left(p_1+p_2+p_3\right)}^7{\left(1-\left(p_1+p_2+p_3\right)\right)}^3$

   $=\left(\begin{array}{c}10\\ 7\end{array}\right){\left(p_1+p_2+p_3\right)}^7{\left(p_4+p_5\right)}^3$

Step 9:Final answer (Part-c)

The value of $P\left(X_1+X_2+X_3=7\right)$ is$\left(\begin{array}{c}10\\ 7\end{array}\right){\left(p_1+p_2+p_3\right)}^7{\left(p_4+p_5\right)}^3$