魅影直播

$P_i,i=1,鈥�,r\underset{1}{\overset{r}{鈭�}}{P}_i=1$

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

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 $k鈭�{\mathrm{鈩�}}_0$, we have that

$P(X=k)={(1鈭�p_1)}^k鈰�p_1$

Therefore, the mean is equal to

$E\left(X\right)=\frac{1鈭�p_1}{p_1}$

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

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

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

$\underset{j=2}{\overset{n}{鈭�}}P(I_j=1)=\underset{j=2}{\overset{n}{鈭�}}\frac{p_j}{p_j+p_1}$

$\text{The required mean is}\underset{j=2}{\overset{n}{鈭�}}\frac{p_j}{p_j+p_1}\text{.}$