Suppose that a random variable x has the following probability density function: f(x) = ( c(1 − x 2 ) if − 1 < x < 1 0, otherwise determine the value of the constant c and find the cumulative distribution function of x. also, compute e[x].

The value of the constant is .

The cumulative distribution function of X is .

The expected value E(X) is 0.

Step-by-step explanation:

1. The function f(x) is a probability density function (pdf) for the continuous random variable X, defined over the set of real numbers, if

We have the following probability density function

We know that for f(x) to be a probability distribution . We integrate f(x) with respect to x, set the result equal to 1 and solve for c.

2. The cumulative distribution function F(x) of a continuous random variable X with density function f(x) is

To find the cumulative distribution function F(x) integrate f(x) from the lower bound of the domain on which f(x) ≠ 0 to x.

3. Let X be a random variable with probability distribution f (x). The mean, or expected value, of X is

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