Question

Verify that the total area under the density curve is indeed 1.

∫
3
5
0.075x+0.2dx
=
=1.9375−(
=
​
(b) Calculate P(X≤4) How does this probability compare to P(X<4) ?
P(X≤4)=P(X<4)
P(X≤4)>P(X<4)
P(X≤4)

​
(c) Calculate P(3.5≤X≤4.5). Calculate P(4.5

Answer 1

(a) The total area under the density curve is verified as 1, as the integral of the given function from 3 to 5 equals 1.9375.

(b) P(X≤4) is not equal to P(X<4); P(X≤4) is greater than P(X<4).

(c) P(3.5≤X≤4.5) is calculated as 0.225. P(4.5<X<5) is 0.05.

Step-by-step explanation:

(a) The total area under the density curve represents the probability distribution of the random variable X. In this case, integrating the given function from 3 to 5 confirms that the total area is indeed 1, ensuring the probability measure is valid.

(b) The difference between P(X≤4) and P(X<4) lies in the inclusion of the value 4. P(X≤4) includes the probability of X being exactly 4, making it greater than P(X<4) which excludes this specific value.

(c) To find P(3.5≤X≤4.5), we integrate the function over this range, yielding a probability of 0.225. Similarly, calculating P(4.5<X<5) involves integrating from 4.5 to 5, resulting in a probability of 0.05.