<Dl> <Dd> Specifically, given X 1 + X 2 = k (\ displaystyle X_ (1) + X_ (2) = k), X 1 ∼ B i n o m (k, λ 1 / (λ 1 + λ 2)) (\ displaystyle \! X_ (1) \ sim \ mathrm (Binom) (k, \ lambda _ (1) / (\ lambda _ (1) + \ lambda _ (2)))). </Dd> <Dd> More generally, if X, X,..., X are independent Poisson random variables with parameters λ, λ,..., λ then <Dl> <Dd> given ∑ j = 1 n X j = k, (\ displaystyle \ sum _ (j = 1) ^ (n) X_ (j) = k,) X i ∼ B i n o m (k, λ i ∑ j = 1 n λ j) (\ displaystyle X_ (i) \ sim \ mathrm (Binom) \ left (k, (\ frac (\ lambda _ (i)) (\ sum _ (j = 1) ^ (n) \ lambda _ (j))) \ right)). In fact, (X i) ∼ M u l t i n o m (k, (λ i ∑ j = 1 n λ j)) (\ displaystyle \ (X_ (i) \) \ sim \ mathrm (Multinom) \ left (k, \ left \ ((\ frac (\ lambda _ (i)) (\ sum _ (j = 1) ^ (n) \ lambda _ (j))) \ right \) \ right)). </Dd> </Dl> </Dd> </Dl> <Dd> Specifically, given X 1 + X 2 = k (\ displaystyle X_ (1) + X_ (2) = k), X 1 ∼ B i n o m (k, λ 1 / (λ 1 + λ 2)) (\ displaystyle \! X_ (1) \ sim \ mathrm (Binom) (k, \ lambda _ (1) / (\ lambda _ (1) + \ lambda _ (2)))). </Dd> <Dd> More generally, if X, X,..., X are independent Poisson random variables with parameters λ, λ,..., λ then <Dl> <Dd> given ∑ j = 1 n X j = k, (\ displaystyle \ sum _ (j = 1) ^ (n) X_ (j) = k,) X i ∼ B i n o m (k, λ i ∑ j = 1 n λ j) (\ displaystyle X_ (i) \ sim \ mathrm (Binom) \ left (k, (\ frac (\ lambda _ (i)) (\ sum _ (j = 1) ^ (n) \ lambda _ (j))) \ right)). In fact, (X i) ∼ M u l t i n o m (k, (λ i ∑ j = 1 n λ j)) (\ displaystyle \ (X_ (i) \) \ sim \ mathrm (Multinom) \ left (k, \ left \ ((\ frac (\ lambda _ (i)) (\ sum _ (j = 1) ^ (n) \ lambda _ (j))) \ right \) \ right)). </Dd> </Dl> </Dd> <Dl> <Dd> given ∑ j = 1 n X j = k, (\ displaystyle \ sum _ (j = 1) ^ (n) X_ (j) = k,) X i ∼ B i n o m (k, λ i ∑ j = 1 n λ j) (\ displaystyle X_ (i) \ sim \ mathrm (Binom) \ left (k, (\ frac (\ lambda _ (i)) (\ sum _ (j = 1) ^ (n) \ lambda _ (j))) \ right)). In fact, (X i) ∼ M u l t i n o m (k, (λ i ∑ j = 1 n λ j)) (\ displaystyle \ (X_ (i) \) \ sim \ mathrm (Multinom) \ left (k, \ left \ ((\ frac (\ lambda _ (i)) (\ sum _ (j = 1) ^ (n) \ lambda _ (j))) \ right \) \ right)). </Dd> </Dl>

Poisson probability density function has ____ parameter(s)