Ryan O ’ Donnell Scribe : Yu Zhao 1 SUM OF RANDOM VARIABLES
نویسنده
چکیده
Let X1, X2, X3, . . . be i.i.d. random variables (Here ”i.i.d.” means ”independent and identically distributed” ), s.t. Pr[Xi = 1] = p, Pr[Xi = 0] = 1 − p. Xi is also called Bernoulli random variable. Let Sn = X1 + · · · + Xn. We will be interested in the random variable Sn which is called Binomial random variable (Sn ∼ B(n, p)). If you toss a coin for n times, and Xi = 1 represents the event that the result is head in the ith turn, then Sn is just the total number of appearance of head in n times. Recall some basic facts on expectation and variance, where Y, Y1, Y2 are random variables. • E[Y1 + Y2] = E[Y1] + E[Y2] • E[Y1Y2] = E[Y1] E[Y2] if random variables Y1 and Y2 are independent (Y1 ⊥ Y2) • E[cY ] = cE[Y ], E[c+ Y ] = c+ E[Y ], where c is a constant • If we denote μ = E[Y ], the variance of Y Var[Y ] = E[(Y − μ)] = E[Y 2 − 2μY + μ] = E[Y ]− 2μE[Y ] + μ = E[Y ]− E[Y ]
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