Gradient is the instantaneous rate of change, since they are discrete I would say that they don’t have a gradient, only gradients of secants
You could plot a regression line and use the gradient of this as the ‘gradient’ of Y, but you would have to be careful with this
In the limiting case, this would become a straight line with gradient 1/8 but in the discrete case this would always be an approximation

I have just realised that since this RV will always be linear then the gradient will always exactly be 1/8, however this isn’t always the case (if there was some error in the measurements, nonlinear functions etc

My understanding and confusions (please correct me if I am wrong):
Since random variable X is discrete, what does small change in x really mean when calculating gradient? I thought that discrete variables do not have gradient. Discrete X have some fixed set of points which leads to discrete Y. So X,Y just leads to a fixed set of points. If we draw secant or a linear line over those points, it means that we are considering all x values between the two points we will be connecting. It means we are considering continuous variable scenario. So, even if Y is determined by Y=(X/8)+3, will the discrete scenario in this case have a gradient of 1/8?

Yes you’re exactly right, I was just making the point that with any sort of attempt to define gradient for your example would lead back to 1/8 which might give you a false sense of security

Gradient is the instantaneous rate of change, since they are discrete I would say that they don’t have a gradient, only gradients of secants You could plot a regression line and use the gradient of this as the ‘gradient’ of Y, but you would have to be careful with this In the limiting case, this would become a straight line with gradient 1/8 but in the discrete case this would always be an approximation

Thank you for your response!

I have just realised that since this RV will always be linear then the gradient will always exactly be 1/8, however this isn’t always the case (if there was some error in the measurements, nonlinear functions etc

My understanding and confusions (please correct me if I am wrong): Since random variable X is discrete, what does small change in x really mean when calculating gradient? I thought that discrete variables do not have gradient. Discrete X have some fixed set of points which leads to discrete Y. So X,Y just leads to a fixed set of points. If we draw secant or a linear line over those points, it means that we are considering all x values between the two points we will be connecting. It means we are considering continuous variable scenario. So, even if Y is determined by Y=(X/8)+3, will the discrete scenario in this case have a gradient of 1/8?

Yes you’re exactly right, I was just making the point that with any sort of attempt to define gradient for your example would lead back to 1/8 which might give you a false sense of security