A r.v. X assumes values 1, 2, 3, n with equal probabilities. (i) If the ratio of Var (X) to E(X) is equal to 4, find n. (ii) What will be n if Var (X) = E(X)?
Answers
Given : A r.v. X assumes values 1, 2, 3, n with equal probabilities.
(i) If the ratio of Var (X) to E(X) is equal to 4
ii) Var (X) = E(X
To find : Value of n
Solution:
Probability are equal
hence probability for each = 1/n
x p xp x²p
1 1/n 1×1/n 1²×1/n
2 1/n 2×1/n 2²×1/n
3 1/n 3×1/n 3²×1/n
....
...
n 1/n n×1/n n²×1/n
mean E(X) = ∑xp = (1/n) (1 + 2 + 3 + .......................+ n) = (1/n) n(n + 1)/2
= (n + 1)/2
∑x²p = (1/n) (1² + 2² + 3² + .......................+ n²) = (1/n) n(n + 1)(2n+1)/6
= (n + 1)(2n+1)/6
Var(X) = ∑x²p - (∑xp)² = (n + 1)(2n+1)/6 - ((n + 1)/2)²
= ((n + 1) / 2) { (2n+1)/3 - (n + 1) / 2}
= ((n + 1) / 12) { (4n+2 - 3n-3}
= ((n + 1) / 12) { (n-1}
Var(X) = (n + 1) (n-1)/12
ratio of Var (X) to E(X) is equal to 4,
=> ( (n + 1) (n-1)/12 ) / ( (n + 1)/2 ) = 4
=> n- 1 = 24
=> n = 25
Var (X) = E(X)
=>( (n + 1) (n-1)/12 ) / ( (n + 1)/2 ) = 1
=> n-1 = 6
=> n = 7
Learn More:
A r.v. X has the given probability distribution, Find the value of k and ...
https://brainly.in/question/6536531
Find k if the following is the p.d.f. of a r.v. X: [tex]f(x) = left{ egin ...
https://brainly.in/question/6536697