Method of moments for an up-day probability

Question

You model each trading day as an independent coin flip: the stock closes up (a $1$ ) with unknown probability $p$ , or not (a $0$ ). Over $250$ days it closed up on $140$ of them.

Use the method of moments to estimate $p$ .

Accepted Answer

Match the mean. For a 0/1 (Bernoulli) variable the theoretical mean is just the success probability: \mathbb{E}[X] = p. Solve. The sample average of a bunch of 0s and 1s is simply the fraction of 1s: \bar{X} = \frac{140}{250} = 0.56. Setting the model mean equal to the sample mean gives \hat p = 0.56. Why the mean is the fraction. Averaging 0s and 1s adds up the ones and divides by the count, which is the same as "share that came up 1." So for a yes/no variable, the method-of-moments estimate of the yes-probability is nothing more than the observed hit rate. Takeaway. For a yes/no outcome the…

Method of moments for an up-day probability

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