Q.7.19 - A First Course in Probability

Question

A certain region is inhabited by r distinct types of a certain species of insect. Each insect caught will, independently of the types of the previous catches, be of type i with probability 

$P_i,i=1,\dots ,r\sum _1^r{P}_i=1$

(a) Compute the mean number of insects that are caught before the first type $1$ catch.

(b) Compute the mean number of types of insects that are caught before the first type $1$ catch.

Step-by-Step Solution

Answer

As per the information, the mean number of insects that are caught and the mean number of types of insects that are caught  are

$\text{(a) The required mean is}\frac{1-p_1}{p_1}\text{.}$

$\text{(b) The required mean is}\sum _{j=2}^n\frac{p_j}{p_j+p_1}\text{.}$

Step 1: Given Information (part a)

$P_i,i=1,\dots ,r\sum _1^r{P}_i=1$

Compute the mean number of insects that are caught before the first type $1$catch. 

Step 2: Explanation (part a)

Define random variable X that counts the number of insects that are caught before the first type 1 catch. Because of the nature of this problem, we have that X has Geometric distribution with a parameter of success p1. So, for $k\in {\mathrm{\mathbb{N}}}_0$, we have that

$P(X=k)={(1-p_1)}^k\cdot p_1$

Therefore, the mean is equal to

$E\left(X\right)=\frac{1-p_1}{p_1}$

Step 3: Final Answer (part a)

$\text{The required mean is}\frac{1-p_1}{p_1}\text{.}$

Step 4: Given Information (part b)

Compute the mean number of types of insects that are caught before the first type $1$ catch.

Step 5: Explanation (part b)

Define indicator random variables $I_{j, }$j$=2$,...,n  that marks if we have caught insect of type j before some insect of type i. Because of the symmetry of this problem, we have that

$P(I_j=1)=\frac{p_j}{p_j+p_1}$

So, the expected number of types of insects that have been caught before the insect of type 1 is 

$\sum _{j=2}^{n}P(I_j=1)=\sum _{j=2}^n\frac{p_j}{p_j+p_1}$

Step 6: Final Answer (part b)

$\text{The required mean is}\sum _{j=2}^n\frac{p_j}{p_j+p_1}\text{.}$