Question

Let X = time (in minutes) that lapses between the ringing of the bell at the end of a lecture and the actual time when the professor ends the lecture. Suppose X has p.d.f.

f(x) = `{(kx^2","      0 ≤ x ≤ 2), (0","         "othenwise"):}`

Then, the probability that the lecture ends within 1 minute of the bell ringing is ______

Options

• `1/2`

• `1/4`

• `1/8`

• `1/16`

Answer 1

Let X = time (in minutes) that lapses between the ringing of the bell at the end of a lecture and the actual time when the professor ends the lecture. Suppose X has p.d.f.

f(x) = `{(kx^2","      0 ≤ x ≤ 2), (0","         "othenwise"):}`

Then, the probability that the lecture ends within 1 minute of the bell ringing is `underline(1/8)`.

Explanation:

Since, f(x) is the p.d.f. of X.

∴ `int_0^2 f(x)dx = 1`

⇒ `int_0^2(kx^2)dx = 1`

⇒ k`[x^3/3]_0^2 = 1`

⇒ `k = 3/8`

∴ Required probability = P(X ≤ 1) = `int_0^1f(x)dx`

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

= `3/8int_0^1x^2dx`

= `3/8[x^3/3]_0^1`

= `3/8(1/3 - 0) = 1/8`