Question

Observe that for a random variable Y that takes on values 0 and​1, the expected value of Y is defined as​ follows: Upper E left parenthesis Upper Y right parenthesis equals 0 times Pr left parenthesis Upper Y equals 0 right parenthesis plus 1 times Pr left parenthesis Upper Y equals 1 right parenthesis ​Now, suppose that X is a Bernoulli random variable with success probability Pr ​(X = 1​) ​= p. Use the information above to answer the following questions. Show that Upper E left parenthesis Upper X cubed right parenthesis equals p. Upper E left parenthesis Upper X cubed right parenthesis ​= ​( nothing times nothing​) ​+ ​( nothing times ​p)​= nothing ​(Use the tool palette on the right to insert superscripts. Enter you answer in the same format as above.​)

Answer 1

Answer:

The expected value of Bernoulli random variable X is p.

$E (X) = 0\times (1-p)+1\times p$

Step-by-step explanation:

A Bernoulli random variable, follows a discrete probability distribution, takes only two values, 0 with probability (1 - p) and 1 with probability p.

Here the random variable X is defined as a Bernoulli random variable.

Then X can assume only two values, i.e. 0 and 1.

It is provided that P (X = 1) = p.

Consider the provided information:

Y = {0, 1}

$E (Y) = 0\times P (Y=0)+1\times P(Y=1)$

The random variable Y is also a Bernoulli random variable.

Using the formula of E (Y) compute the value of E (X) as follows:

$E (X) = 0\times P (X=0)+1\times P(X=1)\\=0\times (1-p)+1\times p\\=0+p\\=p$

Thus, the expected value of Bernoulli random variable X is p.

$E (X) = 0\times (1-p)+1\times p$