IB Mathematics Applications & Interpretation Question Type 3: Calculating with a Linear Combination of Independent Poisson Distributions Exercises

Let $X\sim \text{Po}(3)$ and $Y\sim \text{Po}(5)$ be independent random variables. Find $\text{P}(X+Y=8)$.

Let $X\sim \mathrm{Poisson}({\lambda }_1)$ and $Y\sim \mathrm{Poisson}({\lambda }_2)$ be independent random variables. Determine $\mathrm{Var}(aX+bY)$ in terms of $a,b,{\lambda }_1$, and ${\lambda }_2$.

A factory has two machines producing defects independently with rates $\text{Po}(2)$ and $\text{Po}(1.5)$ per day. Find the probability that fewer than 3 defects occur in total in a single day.

Find the moment generating function $M_Z(t)$ of $Z=2X+3Y$ where $X\sim \text{Po}({\lambda }_1)$ and $Y\sim \text{Po}({\lambda }_2)$ are independent random variables.

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

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

Let $X\sim \text{Po}({\lambda }_1)$ and $Y\sim \text{Po}({\lambda }_2)$ be independent random variables.

Show that $X+Y\sim \text{Po}({\lambda }_1+{\lambda }_2)$.

Independent random variables $X$ and $Y$ are such that $X\sim \text{Po}(4)$ and $Y\sim \text{Po}(6)$.

Show that $\text{P}(X-Y=1)=4{\text{e}}^{-10}{\sum }_{y=0}^{\infty }\frac{24^y}{y!(y+1)!}$.

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

Determine the mean and the variance of $Z=4X-2Y$, given that $X$ and $Y$ are independent random variables with $X\sim \text{Po}(3)$ and $Y\sim \text{Po}(4)$.

Call centre A receives calls according to a Poisson process at a rate of 5 calls per hour, and call centre B at a rate of 3 calls per hour.

Calculate the probability that exactly 16 calls are received in total during a 2-hour period.

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?