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I've upvoted a couple answers that already provide many of the ingredients to the answer. I'll provide what I view as a more direct answer.

Suppose you find a dataset with observations on 2 fields: x (fertilizer) and y (yield) but you don't know exactly how this dataset was obtained. You think of Fisher's experiments and realize that this is probably experimental data where x (amount of fertilizer) was set by the experimenter and after some time the corresponding y (crop yield) was measured. You want to fit the model $y=\beta_0+\beta_1x+\epsilon$.

What would it mean for you to treat x as non-random/fixed?

To treat x as non-random means to assume that the x was set by the experimenter in the precise sense that:

• $E(\epsilon|x)=E(\epsilon)$
• $Var(\epsilon|x)=Var(\epsilon)$

This is what most textbooks mean by a non-random regressor. Not only is x under the experimenter's control but it has been set in a particular way. For example, if the experimenter randomly pulled fertilizer amounts out of a hat, that would meet the above conditions. On the other hand, if the experimenter set fertilizer amounts as a function of plot quality, this would not meet the above conditions.

What would it mean for you to treat x as random?

To treat x as random means to assume that this is observational data where the x was merely observed and not set, which really says that we do not know the probability distribution of x.

Is there any other thing we could assume?

We could assume that x was set by the experimenter in a way that violated one of the above 2 conditions. This is still a non-random regressor in the dictionary sense as Var(x)=0 but this is in conflict with what textbooks mean by "non-random regressor". If the experimenter set fertilizer amounts as a function of plot quality then $E(\epsilon|x)\not=E(\epsilon)$ and even if we further assume that $E(\epsilon)=0$, note that $E(y|x)=\beta_0+\beta_1x+E(\epsilon|x)\not= \beta_0+\beta_1x+E(\epsilon)=\beta_0+\beta_1x$.

I've upvoted a couple answers that already provide many of the ingredients to the answer. I'll provide what I view as a more direct answer.

Suppose you find a dataset with observations on 2 fields: x (fertilizer) and y (yield) but you don't know exactly how this dataset was obtained. You think of Fisher's experiments and realize that this is probably experimental data where x (amount of fertilizer) was set by the experimenter and after some time the corresponding y (crop yield) was measured. You want to fit the model $y=\beta_0+\beta_1x+\epsilon$.

What would it mean for you to treat x as non-random/fixed?

To treat x as non-random means to assume that the x was set by the experimenter in the precise sense that:

• $E(\epsilon|x)=E(\epsilon)$
• $Var(\epsilon|x)=Var(\epsilon)$

This is what most textbooks mean by a non-random regressor. Not only is x under the experimenter's control but it has been set in a particular way. For example, if the experimenter randomly pulled fertilizer amounts out of a hat, that would meet the above conditions. On the other hand, if the experimenter set fertilizer amounts as a function of plot quality, this would not meet the above conditions.

In this setting we would assume $E(\epsilon)=0$ and $Var(\epsilon)=\sigma^2$.

What would it mean for you to treat x as random?

To treat x as random means to assume that this is observational data where the x was merely observed and not set, which really says that we do not know the probability distribution of x.

In this setting we would assume $E(\epsilon|x)=0$ and $Var(\epsilon|x)=\sigma^2$.

Is there any other thing we could assume?

We could assume that x was set by the experimenter in a way that violated one of the above 2 conditions. This is still a non-random regressor in the dictionary sense as Var(x)=0 but this is in conflict with what textbooks mean by "non-random regressor". If the experimenter set fertilizer amounts as a function of plot quality then $E(\epsilon|x)\not=E(\epsilon)$ and even if we further assume that $E(\epsilon)=0$, note that $E(y|x)=\beta_0+\beta_1x+E(\epsilon|x)\not= \beta_0+\beta_1x+E(\epsilon)=\beta_0+\beta_1x$.

I've upvoted a couple answers that already provide many of the ingredients to the answer. I'll provide what I view as a more direct answer.

Suppose you find a dataset with observations on 2 fields: x (fertilizer) and y (yield) but you don't know exactly how this dataset was obtained. You think of Fisher's experiments and realize that this is probably experimental data where x (amount of fertilizer) was set by the experimenter and after some time the corresponding y (crop yield) was measured. You want to fit the model $y=\beta_0+\beta_1x+\epsilon$.

What would it mean for you to treat x as non-random/fixed?

To treat x as non-random means to assume that the x was set by the experimenter in the precise sense that:

• $E(\epsilon|x)=E(\epsilon)$
• $Var(\epsilon|x)=Var(\epsilon)$

For example, if the experimented randomly pulled fertilizer amounts out of a hat, that would meet the above conditions. On the other hand, if the experimenter set fertilizer amounts as a function of plot quality, this would not meet the above conditions.

What would it mean for you to treat x as random?

To treat x as random means to assume that the x was set by the experimenter in a way that violated one of the above 2 conditions (note this is still an experimental setting with x under the experimenter's control) or to assume that this is observational data where the x was altogether not in the experimenter's control.

I've upvoted a couple answers that already provide many of the ingredients to the answer. I'll provide what I view as a more direct answer.

Suppose you find a dataset with observations on 2 fields: x (fertilizer) and y (yield) but you don't know exactly how this dataset was obtained. You think of Fisher's experiments and realize that this is probably experimental data where x (amount of fertilizer) was set by the experimenter and after some time the corresponding y (crop yield) was measured. You want to fit the model $y=\beta_0+\beta_1x+\epsilon$.

What would it mean for you to treat x as non-random/fixed?

To treat x as non-random means to assume that the x was set by the experimenter in the precise sense that:

• $E(\epsilon|x)=E(\epsilon)$
• $Var(\epsilon|x)=Var(\epsilon)$

This is what most textbooks mean by a non-random regressor. Not only is x under the experimenter's control but it has been set in a particular way. For example, if the experimenter randomly pulled fertilizer amounts out of a hat, that would meet the above conditions. On the other hand, if the experimenter set fertilizer amounts as a function of plot quality, this would not meet the above conditions.

What would it mean for you to treat x as random?

To treat x as random means to assume that this is observational data where the x was merely observed and not set, which really says that we do not know the probability distribution of x.

Is there any other thing we could assume?

We could assume that x was set by the experimenter in a way that violated one of the above 2 conditions. This is still a non-random regressor in the dictionary sense as Var(x)=0 but this is in conflict with what textbooks mean by "non-random regressor". If the experimenter set fertilizer amounts as a function of plot quality then $E(\epsilon|x)\not=E(\epsilon)$ and even if we further assume that $E(\epsilon)=0$, note that $E(y|x)=\beta_0+\beta_1x+E(\epsilon|x)\not= \beta_0+\beta_1x+E(\epsilon)=\beta_0+\beta_1x$.

I've upvoted a couple answers that already provide many of the ingredients to the answer. I'll provide what I view as a more direct answer.

Suppose you find a dataset with observations on 2 fields: x (fertilizer) and y (yield) but you don't know exactly how this dataset was obtained. You think of Fisher's experiments and realize that this is probably experimental data where x (amount of fertilizer) was set by the experimenter and after some time the corresponding y (crop yield) was measured. You want to fit the model $y=\beta_0+\beta_1x+\epsilon$.

What would it mean for you to treat x as non-random/fixed?

To treat x as non-random means to assume that the x was set by the experimenter in the precise sense that:

• $E(\epsilon|x)=E(\epsilon)$
• $Var(\epsilon|x)=Var(\epsilon)$

What would it mean for you to treat x as random?

To treat x as random means to assume that the x was set by the experimenter in a way that violated one of the above 2 conditions (note this is still an experimental setting with x under the experimenter's control) or to assume that this is observational data where the x was altogether not in the experimenter's control.

I've upvoted a couple answers that already provide many of the ingredients to the answer. I'll provide what I view as a more direct answer.

Suppose you find a dataset with observations on 2 fields: x (fertilizer) and y (yield) but you don't know exactly how this dataset was obtained. You think of Fisher's experiments and realize that this is probably experimental data where x (amount of fertilizer) was set by the experimenter and after some time the corresponding y (crop yield) was measured. You want to fit the model $y=\beta_0+\beta_1x+\epsilon$.

What would it mean for you to treat x as non-random/fixed?

To treat x as non-random means to assume that the x was set by the experimenter in the precise sense that:

• $E(\epsilon|x)=E(\epsilon)$
• $Var(\epsilon|x)=Var(\epsilon)$

For example, if the experimented randomly pulled fertilizer amounts out of a hat, that would meet the above conditions. On the other hand, if the experimenter set fertilizer amounts as a function of plot quality, this would not meet the above conditions.

What would it mean for you to treat x as random?

To treat x as random means to assume that the x was set by the experimenter in a way that violated one of the above 2 conditions (note this is still an experimental setting with x under the experimenter's control) or to assume that this is observational data where the x was altogether not in the experimenter's control.