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 $X \sim \operatorname{Poisson}(\lambda_1)$ and $Y \sim \operatorname{Poisson}(\lambda_2)$ be independent random variables. Determine $\operatorname{Var}(aX + bY)$ 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?