Question about gamma distribution

[semidevil]

so I don't even know where to start and how to approach this. suppose that a set of measurements y1, y2,...y100 were taken from a gamma pdf where E(y)= 1.5, and var(y) = .75. how many y(i's) would I expect to find in the interval (1.0, 2.5).

I have absolutely no idea where to start given my information.

so I know that gamma pdf = lamda^r / (r - 1)! * ye^(-lamda*y), for y > 0.

and i know the expected value formula and variance formula, e(y) = r/lamda and variance(y) = r/lamda^2.

but so what? what do I do?

 

[TenaliRaman]

100*P(1<=y<=2.5)

-- AI

 

[tacman]

Don't worry, it can be overwhelming to approach a new concept like the gamma distribution. Let's break down the problem step by step.

First, we know that the gamma distribution is a continuous probability distribution that is commonly used to model wait times, such as the time between customer arrivals or the lifespan of a product.

In this case, we are given the expected value and variance of the measurements, which are both properties of the gamma distribution. The expected value, denoted as E(y), is the mean or average of the measurements, and the variance, denoted as var(y), measures the spread or variability of the measurements.

Next, we are given an interval, (1.0, 2.5), and we are asked to find the number of measurements that fall within this interval. This is known as finding the probability of a range of values in a continuous distribution.

To solve this problem, we can use the probability density function (pdf) of the gamma distribution. This function, as you mentioned, is given by gamma pdf = lambda^r / (r-1)! * y^(-lambda*y), for y > 0.

We can use the properties of the gamma distribution to find the value of lambda, which is the shape parameter, and r, which is the scale parameter. From the information given, we can set up the following equations:

E(y) = r/lambda = 1.5
var(y) = r/lambda^2 = 0.75

Solving for lambda and r, we get lambda = 2 and r = 3.

Now, we can plug these values into the gamma pdf and integrate over the given interval to find the probability of the measurements falling within (1.0, 2.5). This will give us the expected number of measurements in this interval.

In summary, to solve this problem, we need to use the properties of the gamma distribution, specifically the expected value and variance, to find the shape and scale parameters. Then, we can use the gamma pdf to find the probability of the measurements falling within the given interval. I hope this helps!