# Poisson Distribution Mean & SD: Solving for Y

• Mo
In summary: In this case, Y has a mean of 17 and a standard deviation of 4sqrt(4) = 8.In summary, the conversation discusses finding the mean and standard deviation of a random variable Y defined by Y = 4X + 1, where X has a Poisson distribution with a mean of 4. The mean of Y is the same as the mean of (4X + 1), and the standard deviation is 8. The variance of Y is 16 times the variance of X.
Mo
I am attempting a past paper question from school, i don't have the answer (and it doesn't look like i will anytime soon!)

The question:
"The Random variable X has a poisson distribution with mean 4. The random variable Y is defined by

Y = 4X + 1

Find the mean and standard deviation of Y"

So .. where do i begin?

I understand that the mean and variance of a poisson distribution is lambda.I know that the square root of the variance is the SD.They are telling us that this R.V X has a mean and variance of 4 right?

Am i right in thinking that the mean of Y is the same as the mean of (4X + 1)?

E(4X + 1) = 4

or is it ...

4E(X) + 1 where E(X) is 4?

Help is very much appreciated!

Regards,
Mo

PS: Stats is not something that i understand all that easy

The 2nd : E(4X+1)=4*E(X)+1...and then i think it's like :

var(Y)=E(Y^2)-E(Y)^2=E(16X^2+8X+1)-(4*E(X)+1)^2=16(E(X^2)-E(X)^2)=16*var(X)...but I'm not sure

Note to Kleinwolf: The variance calculation is correct. In general, adding a constant leaves the variance unchanged, while multiplying by a constant changes the variance by multiplying by the constant squared.

## 1. What is the Poisson Distribution?

The Poisson Distribution is a discrete probability distribution that is used to model the number of occurrences of a specific event within a fixed interval of time or space. It is often used in situations where the probability of the event occurring is small and the events are independent of each other.

## 2. How is the mean of the Poisson Distribution calculated?

The mean of the Poisson Distribution is calculated by multiplying the rate of occurrence (λ) by the time or space interval (t). This can be represented by the formula μ = λt.

## 3. What is the significance of the mean in the Poisson Distribution?

The mean in the Poisson Distribution represents the average number of events that are expected to occur within the given interval. It is also used as a measure of central tendency for the distribution.

## 4. What is the standard deviation in the Poisson Distribution?

The standard deviation in the Poisson Distribution is calculated using the square root of the mean (σ = √μ). It represents the measure of variability or spread of the distribution around the mean.

## 5. How is the Poisson Distribution used to solve for Y?

To solve for Y, which represents the number of events occurring within a given interval, we can use the Poisson Probability formula which is P(Y; μ) = (e^-μ * μ^Y) / Y!. This formula calculates the probability of Y events occurring, given the mean (μ) of the distribution.

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