Final answer:
The cumulative distribution function (CDF) is found by integrating the probability density function (PDF) over the desired range. Expected value (E[X]) is calculated by integrating x times the PDF, and variance is calculated by subtracting the squared expected value from the expected value of the squared variable. To find P(X > 0.2), we use the complement rule.
Step-by-step explanation:
(a) Cumulative Distribution Function (CDF):
The cumulative distribution function (CDF), denoted as F(x), of a continuous random variable X is defined as the probability that X takes on a value less than or equal to x. In this case, we have:
F(x) = ∫[0,x] f(t) dt
The limits of integration are from 0 to x, and f(t) is the probability density function (PDF) of X. Since the PDF is given by f(x) = 3/2(1−x^2) for 0≤x≤1 and 0 otherwise, we can substitute this into the integral:
F(x) = ∫[0,x] 3/2(1−t^2) dt
To find the CDF, we evaluate this integral:
F(x) = ∫[0,x] 3/2(1−t^2) dt = [3/2(t−t^3/3)]|0^x = 3/2(x−x^3/3)
(b) Expected Value (E[X]) and Variance:
The expected value or mean of a continuous random variable X, denoted as E[X], can be calculated using the formula:
E[X] = ∫x f(x) dx
Again, using the PDF f(x) = 3/2(1−x^2), we substitute this into the integral:
E[X] = ∫x 3/2(1−x^2) dx = 3/2(x−x^3/3)
To find the variance of X, denoted as Var(X), we use the formula:
Var(X) = E[X^2] − (E[X])^2
We need to evaluate ∫x^2 f(x) dx to find E[X^2].
After evaluating the integral, we can calculate the variance.
(c) To find P(X > 0.2), we need to use the complement rule:
P(X > 0.2) = 1 − P(X ≤ 0.2)
We can substitute the CDF function from part (a) into this expression to find the probability.