The diameter of an electric cable. Say X, is assumed to be a continuous random variable
with p.d.f.: f(x) = 6x(1 – x), 0 x 1.
i) Check that f(x) is p.d.f.
ii) Determine a number b such that P(X < b) = P(X > b).
Answers
Answer:
Given : The diameter of an electric cable, Say X, is assumed to be continuous random variable with p.d.f f (x)= 6x(1-x), 0 ≤ x ≤ 1
To find : Show that f(x) is a p.d.f. (probability density function )
Solution:
f(x) = 6x(1 - x) 0 ≤ x ≤ 1
Function f(x) is a probability density function in range a to b
if \int\limits^b_a {f(x)} \, dx = 1
a
∫
b
f(x)dx=1
f(x) = 6x(1 - x)
f(x) = 6x - 6x²
\int\limits^1_0 {(6x - 6x^2)} \, dx
0
∫
1
(6x−6x
2
)dx
= [ \frac{6x^2}{2} - \frac{6x^3}{3} ] ^1_0[
2
6x
2
−
3
6x
3
]
0
1
= [ 3x^2 - 2x^3 ] ^1_0[3x
2
−2x
3
]
0
1
= 3(1)² - 2(1)³ - ( 0 - 0)
= 3 - 2 - 0
= 1
\int\limits^1_0 {(6x - 6x^2)} \, dx
0
∫
1
(6x−6x
2
)dx = 1
Hence f(x) = 6x(1 - x) 0 ≤ x ≤ 1 is pdf ( probability density function )
Answer:
The diameter of an electric cable. Say X, is assumed to be a continuous random variable
with p.d.f.: f(x) = 6x(1 – x), 0 x 1.
i) Check that f(x) is p.d.f.
ii) Determine a number b such that P(X < b) = P(X > b).