Question Type 3: Calculating with a linear combination of independent poisson distributions Exercises

Let $X\sim Poisson(3)$ and $Y\sim Poisson(5)$ be independent. Find $P(X+Y=8)$.

Let $X\sim Poisson(\lambda_1)$ and $Y\sim Poisson(\lambda_2)$ be independent. Determine $Var(aX+bY)$ in terms of $a,b,\lambda_1,\lambda_2$.

A factory has two machines producing defects independently with rates $Poisson(2)$ and $Poisson(1.5)$ per day. What is the probability that fewer than 3 defects occur in total in a single day?

A call centre A receives calls according to a Poisson process at rate 5 calls per hour, and centre B at rate 3 calls per hour. What is the probability that exactly 16 calls are received in total during a 2-hour period?

If $X\sim Poisson(\lambda)$ satisfies $P(X=0)=0.05$, find the value of $\lambda$.

Compute the mean and variance of $Z=4X-2Y$ given independent $X\sim Poisson(3)$ and $Y\sim Poisson(4)$.

If $X\sim Poisson(\lambda_1)$ and $Y\sim Poisson(\lambda_2)$ are independent, show that $X+Y\sim Poisson(\lambda_1+\lambda_2)$.

Given independent random variables $X\sim Poisson(x+5)$ and $Y\sim Poisson(x^2-6)$, and $E(6X+Y)=16$, find the value of $x$.

Restaurant X receives on average 4 orders per minute and restaurant Y receives 2 orders per minute. Assuming independence, what is the probability that in a 10-minute period they receive more than 70 orders in total?

Find the moment generating function $M_Z(t)$ of $Z=2X+3Y$ where $X\sim Poisson(\lambda_1)$, $Y\sim Poisson(\lambda_2)$ are independent.

Given independent $X\sim Poisson(2)$ and $Y\sim Poisson(3)$, find the conditional distribution of $X$ given $X+Y=5$.

For independent $X\sim Poisson(4)$ and $Y\sim Poisson(6)$, find $P(X-Y=1)$.