Suppose the income (in tens of thousands of pesos) of employees at University
X can be approximated by a continuous distribution with density
f(x) =
(
2x2; if x � 2
0; if x < 2
Find the probability that a randomly chosen employee has an income between P30,000
and P50,000.
2. 9 pts.
(a) Determine the value of c so that the following function is a probability density
function (pdf):
f(x) =
8>>><
>>>:
15
64
+
x
64
; if 2 � x � 0
3
8
+ cx; if 0 < x � 3
0; otherwise
(b) Find P(1 � X � 1).
(c) Find F(x), the cumulative density function.
3. 12 pts. Let X be a standard normal random variable. Find the following:
(a) P(0 � X � 1:42)
(b) P(0:73 � X � 0)
(c) P(1:37 � X � 2:01)
(d) P(X � 1:13)
4. 5 pts. A fair die is tossed 180 times. Find
To find the probability that a randomly chosen employee has an income between ₱30,000 and ₱50,000, we need to calculate the area under the density function curve between these two points.
Given that the density function is defined as:
f(x) =
2x - 2, if x >= 2
0, if x < 2
We can integrate the function from 30,000 to 50,000 to find the probability:
P(30,000 <= X <= 50,000) = ∫(30,000 to 50,000) f(x) dx
∫(30,000 to 50,000) (2x - 2) dx
To solve this integral, we can split it into two integrals based on the piecewise definition of the density function.
First, integrate from 30,000 to 40,000:
∫(30,000 to 40,000) (2x - 2) dx = ∫(30,000 to 40,000) (2x) dx - ∫(30,000 to 40,000) (2) dx
Integrating 2x:
= [x²] (30,000 to 40,000) - 2 * [x] (30,000 to 40,000)
Substituting the limits:
= (40,000² - 30,000²) - 2 * (40,000 - 30,000)
Next, integrate from 40,000 to 50,000:
∫(40,000 to 50,000) (2x - 2) dx = ∫(40,000 to 50,000) (2x) dx - ∫(40,000 to 50,000) (2) dx
Integrating 2x:
= [x²] (40,000 to 50,000) - 2 * [x] (40,000 to 50,000)
Substituting the limits:
= (50,000² - 40,000²) - 2 * (50,000 - 40,000)
Finally, sum up the results from both integrals to get the probability of income between 30,000 and 50,000.