Question

If `f(x){:(= kx^2(1 - x)",", "for"  0 < x < 1),(= 0",", "otherwise"","):}`

is the probability distribution function of a random variable X, then the value of k is ______.

पर्याय

• 12

• 10

• −9

• −12

Answer 1

If `f(x){:(= kx^2(1 - x)",", "for"  0 < x < 1),(= 0",", "otherwise"","):}`

is the probability distribution function of a random variable X, then the value of k is 12.

Explanation:

Step 1: Understanding the probability distribution function (PDF):

`int_0^1 f(x) dx = 1`

Step 2: Set up the integral:

`int_0^1 kx^2(1 - x) dx = 1`

Factor out the constant k:

`k int_0^1 x^2 (1 - x) dx = 1`

Step 3: Expand the function inside the integral:

= `k int_0^1 (x^2 - x^3)dx`

= `k(int_0^1 x^2 dx - int_0^1 x^3 dx)`

Step 4: Solve each term of the integral:

`int_0^1 x^2 dx = [(x^3)/(3)]_0^1 = 1/3`

`int_0^1 x^3 dx = [(x^4)/(4)]_0^1 = 1/4`

Step 5: Substitute the results back:

`k = (1/3 - 1/4)`

= `k (4/12 - 3/12)`

= `k (1/12)`

Step 6: Equate to 1 and solve for k:

`k (1/12) = 1`

k = 12