(b) A random variable gives measurements X
between 'O' and '1' with a probability function
f(x) = 12x3 - 21x2 +10x and 0, 0<xsi
(i) Find PX
Find P(x * ) and P(x)
1
(ii) Find a number k such that P(X <k)
2
Answers
Answer:
i) P(X<=1/2) = 9/16 and P(X>1/2) = 7/16
Step-by-step explanation:
Given data:
f(x) = 12x^3 - 21x^2 + 10x ; 0<=x<=1
i)
Now,
P(x<=1/2) = lim 0 to 1/2 [∫(12x^3 - 21x^2 + 10x)dx]
= lim 0 to 1/2 [3x^4-7x^3+5x^2]
= 3{1/16-0} - 7{1/8-0} + 5{1/4-0}
= (3 - 14 + 20)/16
= 9/16.
And,
P(x>1/2) = lim 1/2 to 1 [∫(12x^3 - 21x^2 + 10x)dx]
= lim 1/2 to 1 [3x^4-7x^3+5x^2]
= 3{1-1/16} - 7{1-1/8} + 5{1-1/4}
= 45/16 - 49/8 + 15/4
= (45 - 98 + 60)/16
= 7/16.
ii)
P(x<=k) = 1/2 (given)
=> lim 0 to k [∫f(x)dx] = 1/2
=> lim 0 to k [∫(12x^3 - 21x^2 + 10x)dx] = 1/2
=> lim 0 to k [3x^4-7x^3+5x^2] = 1/2
=> 3k^4 - 7k^3 + 5k^2 = 1/2
=> k^2 (3k^2 - 7k + 5) = 1/2
=> k = 0.45175, k = -2.6587.