Advertisements
Advertisements
Question
Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.
f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
compute P(X < – 0.5 or X > 0.5)
Solution
Given that f(x) represents a p.d.f. of r.v. X.
∴ `int_-2^2 f(x)*dx` = 1
∴ `int_-2^2 "k"(4 - x^2)*dx` = 1
∴ `"k"[4x - x^3/3]_-2^2` = 1
∴ `"k"[(8 - 8/3) - (-8 + 8/3)]` = 1
∴ `"k"(16/3 + 16/3)` = 1
∴ `"k"(32/3)` = 1
∴ k = `(3)/(32)`
F(x) = `int_-2^2 f(x)*dx`
= `int_-2^2"k"(4 - x^2)*dx`
= `(3)/(32)[4x - x^3/3]_-2^2`
= `(3)/(32)[4x - x^3/3 + 8 - 8/3]`
∴ F(x) = `(3)/(32)[4x - x^3/3 + 16/3]`
P(X < – 0.5 or X > 0.5)
= 1 – P(– 0.5 ≤ X ≤ 0.5)
= 1 – [F(0.5) – F(– 0.5)]
= `1 - {3/32[4(0.5) - (0.5)^3/3 + 16/3] - 3/32[4(-0.5) - (0.5)^3/3 + 16/3]`
= `1 - 3/32(2 - 1/24 + 16/3 + 2 - 1/24 - 16/3)`
= `1 - 3/32(4 - 1/12)`
= `1 - (3)/(32) xx (47)/(12)`
= `1 - (47)/(128)`
= `(81)/(128)`.
APPEARS IN
RELATED QUESTIONS
The time (in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable taking values between 25 and 35 minutes with p.d.f
`f(x) = {{:(1/10",", 25 ≤ x ≤ 35),(0",", "otherwise"):}`
What is the probability that preparation time exceeds 33 minutes? Also, find the c.d.f. of X.
Verify which of the following is p.d.f. of r.v. X:
f(x) = sin x, for 0 ≤ x ≤ `π/2`
Check whether the following is a p.d.f.
f(x) = `{(x, "for" 0 ≤ x ≤ 1),(2 - x, "for" 1 < x ≤ 2.):}`
The following is the p.d.f. of a r.v. X.
f(x) = `{(x/(8), "for" 0 < x < 4),(0, "otherwise."):}`
Find P(X > 2)
Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.
f(x) = `{(0.5x, "for" 0 ≤ x ≤ 2),(0, "otherwise".):}`
Calculate : P(X ≤ 1)
Let X be the amount of time for which a book is taken out of library by a randomly selected student and suppose that X has p.d.f.
f(x) = `{(0.5x, "for" 0 ≤ x ≤ 2),(0, "otherwise".):}`
Calculate : P(X ≥ 1.5)
Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.
f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
compute P(X > 0)
Suppose error involved in making a certain measurement is a continuous r. v. X with p.d.f.
f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
compute P(–1 < X < 1)
Following is the p. d. f. of a continuous r.v. X.
f(x) = `{(x/8, "for" 0 < x < 4),(0, "otherwise".):}`
Find expression for the c.d.f. of X.
Following is the p. d. f. of a continuous r.v. X.
f(x) = `{(x/8, "for" 0 < x < 4),(0, "otherwise".):}`
Find F(x) at x = 0.5, 1.7 and 5.
The p.d.f. of a continuous r.v. X is
f(x) = `{((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}`
Determine the c.d.f. of X and hence find P(X < 1)
The p.d.f. of a continuous r.v. X is
f(x) = `{((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}`
Determine the c.d.f. of X and hence find P(1 < X < 2)
Choose the correct alternative :
If p.m.f. of r.v.X is given below.
| x | 0 | 1 | 2 |
| P(x) | q2 | 2pq | p2 |
Then Var(X) = _______
Solve the following problem :
Suppose error involved in making a certain measurement is a continuous r.v.X with p.d.f.
f(x) = `{("k"(4 - x^2), "for" -2 ≤ x ≤ 2),(0, "otherwise".):}`
Compute P(–1 < X < 1)
For the following probability density function of a random variable X, find P(X < 1).
`{:(f(x) = (x + 2)/18,";" "for" -2 < x < 4),( = 0,"," "otherwise"):}`
Find k, if the following function is p.d.f. of r.v.X:
f(x) = `{:(kx^2(1 - x)",", "for" 0 < x < 1),(0",", "otherwise"):}`
Find the c.d.f. F(x) associated with the following p.d.f. f(x)
f(x) = `{{:(3(1 - 2x^2)",", 0 < x < 1),(0",", "otherwise"):}`
Find `P(1/4 < x < 1/3)` by using p.d.f. and c.d.f.
