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) =
(
2x􀀀2; 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 the probability P that the face 6 will
appear between 29 and 32 times inclusive.
1

To find the probability that a randomly chosen employee at University X has an income between P30,000 and P50,000, we need to calculate the area under the density function curve between x = 3 and x = 5 (since each unit on the x-axis represents tens of thousands of pesos).

The density function f(x) = 2x - 2 for x >= 2 and f(x) = 0 for x < 2 represents the income distribution.

To find the probability, we integrate the density function over the range from 3 to 5:

P(3 <= X <= 5) = ∫(from 3 to 5) (2x - 2) dx

= [x^2 - 2x] (from 3 to 5) (using the antiderivative of 2x - 2)

= [(5^2 - 2*5) - (3^2 - 2*3)]

= [25 - 10 - 9 + 6]

= 12

Therefore, the probability that a randomly chosen employee at University X has an income between P30,000 and P50,000 is 12%.

Note: It's important to mention that the density function must integrate to 1 over its entire range for it to be a valid probability density function.