The random variable X has the following density function:
f(x) = { x^2 if 0 < x < 3
{ 0 otherwise
a) Find the distribution function F(x) of X
b) Draw the distribution function
c) Calculate the following probabilities:
P(X > 1.5) =
P(1/2 ≤ X < 3/2) =
P(-2 ≤ X < 1) =
d) What is E(X) and Var(X)? Do this calculation only as far as you can without a calculator
well I guess if it looks like x^2 and the integral from 0 to 3 had to be all of it then it looks like
integral ax^2 dx from 0 to 3 = 1
a (3^3)/3 = 1
9 a = 1
a = (1/9)
so distribution is F(x) = x^2/9 defined only where x is between 0 and 3
now for example between 1.5 and 3 (your first domain)
integral (1/9) x^2 dx = (1/9)[ 3^3/3 - 1.5^3/3] = (1/27)[27-3.375 ] = .875
so probability that it is between 1.5 and 3 is 7/8
a) The distribution function F(x) of a random variable X can be found by integrating the density function f(x) from negative infinity to x. In this case, we have:
F(x) = ∫[0 to x] x^2 dx
To find F(x), we need to integrate the function x^2 with respect to x from 0 to x. Integrating x^2, we get the antiderivative x^3/3. Applying the limits, we have:
F(x) = ∫[0 to x] x^2 dx
= [x^3/3] [0 to x]
= (x^3/3) - (0^3/3)
= x^3/3
b) To draw the distribution function, we plot F(x) = x^3/3 on the x-axis for 0 ≤ x ≤ 3. The graph will start from the origin (0,0) and end at the point (3,1) since the density function is defined for 0 < x < 3.
c) To calculate the probabilities, we use the distribution function F(x) calculated in part (a).
P(X > 1.5) = 1 - P(X ≤ 1.5)
= 1 - F(1.5)
= 1 - (1.5^3/3)
P(1/2 ≤ X < 3/2) = F(3/2) - F(1/2)
= (3/2)^3/3 - (1/2)^3/3
P(-2 ≤ X < 1) = F(1) - F(-2)
= 1^3/3 - (-2^3/3)
d) To calculate E(X) (the expected value) and Var(X) (the variance), we use the formulas:
E(X) = ∫[0 to ∞] x * f(x) dx
Var(X) = E(X^2) - [E(X)]^2
To calculate E(X), we multiply the random variable x by its probability density function f(x) and integrate from 0 to 3:
E(X) = ∫[0 to 3] x * x^2 dx
To calculate Var(X), we need to calculate E(X^2) first. We square the random variable x and multiply by its probability density function f(x), and integrate from 0 to 3:
E(X^2) = ∫[0 to 3] x^2 * x^2 dx
Then we can calculate Var(X) using the formula mentioned above.