2. Given the cumulative distribution function (cdf)
F(x)=(0 x<4
(x-4)/4 4 <= x <8
1 x >=8
2.1 Using the cdf, compute P(5<X<6).
2.2 What is the pdf of X?
2.3 Using the pdf, compute P(X<7).
2.4 What is E(X)?
2.5 What is V(X)?
To answer these questions, we need to understand the concepts of cumulative distribution function (CDF) and probability density function (PDF).
1. CDF (F(x)): The cumulative distribution function F(x) gives the probability that a random variable X takes on a value less than or equal to x.
2. PDF (f(x)): The probability density function f(x) gives the derivative of the CDF, i.e., the rate at which the CDF increases at each point.
Now, let's address each question:
2.1 To compute P(5 < X < 6), we can use the CDF.
The CDF is defined as follows:
F(x) = 0, x < 4,
(x - 4)/4, 4 <= x < 8,
1, x >= 8.
P(5 < X < 6) = F(6) - F(5)
= ((6 - 4)/4) - ((5 - 4)/4) (using the CDF values at x=6 and x=5)
= 1/2 - 1/4
= 1/4
Therefore, P(5 < X < 6) is 1/4.
2.2 To find the PDF of X, we can take the derivative of the CDF.
For 4 <= x < 8, the CDF is (x - 4)/4. Taking the derivative with respect to x gives the PDF.
f(x) = d/dx (x - 4)/4
= 1/4
Therefore, the PDF of X is f(x) = 1/4 for 4 <= x < 8.
2.3 To compute P(X < 7), we can also use the CDF.
P(X < 7) = F(7)
= (7 - 4)/4 (using the CDF value at x=7)
= 3/4
Therefore, P(X < 7) is 3/4.
2.4 To find the expected value E(X), we can integrate the product of x and the PDF.
E(X) = ∫(x * f(x)) dx, for the range where f(x) is non-zero
E(X) = ∫(x * 1/4) dx, for 4 <= x < 8
= (1/4) * ∫(x) dx, for 4 <= x < 8
= (1/4) * (x^2/2) ∣ 4 to 8
= (1/4) * (8^2/2 - 4^2/2)
= (1/4) * (32 - 8)
= (1/4) * 24
= 6
Therefore, E(X) is 6.
2.5 To find the variance V(X), we can use the formula V(X) = E(X^2) - (E(X))^2, where E(X) is the mean.
E(X^2) can be calculated by integrating x^2 times the PDF.
E(X^2) = ∫(x^2 * f(x)) dx, for the range where f(x) is non-zero
E(X^2) = ∫(x^2 * 1/4) dx, for 4 <= x < 8
= (1/4) * ∫(x^2) dx, for 4 <= x < 8
= (1/4) * (x^3/3) ∣ 4 to 8
= (1/4) * ((8^3/3) - (4^3/3))
= (1/4) * (512/3 - 64/3)
= (1/4) * (448/3)
= 112/3
Now, we can calculate V(X):
V(X) = E(X^2) - (E(X))^2
= 112/3 - (6)^2
= 112/3 - 36/1
= 112/3 - 108/3
= 4/3
Therefore, V(X) is 4/3.