2 years ago

an item is regularly priced at $30. it is on sale for 30% off the regular price. what is the sale price?

Mathematics

1 answer:

Alex Ar [27]2 years ago

8 0

I would say the new price is $21

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We have n = 100 many random variables xi 's, where the xi 's are independent and identically distributed bernoulli random variab

Alex777 [14]

Recall that for a random variable $X$ following a Bernoulli distribution $\mathrm{Ber}(p)$ , we have the moment-generating function (MGF)

$M_X(t)=(1-p+pe^t)$

and also recall that the MGF of a sum of i.i.d. random variables is the product of the MGFs of each distribution:

$M_{X_1+\cdots+X_n}(t)=M_{X_1}(t)\times\cdots\times M_{X_n}(t)$

So for a sum of Bernoulli-distributed i.i.d. random variables $X_i$ , we have

$M_{\sum\limits_{i=1}^nX_i}(t)=\displaystyle\prod_{i=1}^n(1-p+pe^t)=(1-p+pe^t)^n$

which is the MGF of the binomial distribution $\mathcal B(n,p)$ . (Indeed, the Bernoulli distribution is identical to the binomial distribution when $n=1$ .)