Question

1. If Y~Pois(λ), simplify the ratio of probabilities: P(Y=k+1)/P(Y = k)​

Answer 1

Let, Y~Pois(λ):

If
`Y` is a random variable that is distributed according to the Poisson distribution with parameter λ, then the probability mass function of
`Y` is given by:

• 
  `P(Y=k)=(e^(-lambda).(lambda)^k)/(k!)`
• 
  `P(Y=k+1)=(e^(-lambda).(lambda)^(k+1))/((k+1)!)`

Simplification;

• 
  `((e^(-lambda).(lambda)^(k+1))/((k+1)!))/((e^(-lambda).(lambda)^(k))/(k!))`

• 
  `=(e^(-lambda).(lambda)^(k+1))/((k+1).k!) (k!)/(e^(-lambda).(lambda)^(k))`

• 
  `=((lambda)^(k+1))/((k+1)) (1)/((lambda)^(k))`

• 
  `=((lambda)^(k).(lambda))/((k+1)) (1)/((lambda)^(k))`

• 
  `=((lambda))/((k+1)) .(1)/(1) =(lambda)/(k+1)`