Advertisements
Advertisements
Question
Find k if the following function represents the p. d. f. of a r. v. X.
f(x) = `{(kx, "for" 0 < x < 2),(0, "otherwise."):}`
Also find `"P"[1/4 < "X" < 1/2]`
Solution
Given that f(x) represents p.d.f of r.v. X.
∴ `int_0^2f(x)*dx` = 1
∴ `int_0^2"k"x*dx` = 1
∴ `"k" int_0^2 x*dx` = 1
∴ `"k"/(2)[x^2]_0^2` = 1
∴ `"k"/(2)[4 - 0]` = 1
∴ `"k"/(2)[4]` = 1
∴ k = `(1)/(2)`
`"P"[1/4 < "X" < 1/2] = int_(1/4)^(1/2)f(x)*dx`
= `int_(1/4)^(1/2) x/(2)*dx`
= `(1)/(2) int_(1/4)^(1/2) x*dx`
= `(1)/(4)[x^2]_(1/4)^(1/2)`
= `(1)/(4)[1/4- 1/16]`
= `(1)/(4)[(4 - 1)/16]`
= `(3)/(64)`.
APPEARS IN
RELATED QUESTIONS
State if the following is not the probability mass function of a random variable. Give reasons for your answer.
| X | 0 | -1 | -2 |
| P(X) | 0.3 | 0.4 | 0.3 |
A random variable X has the following probability distribution:
| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P(X) | 0 | k | 2k | 2k | 3k | k2 | 2k2 | 7k2 + k |
Determine:
- k
- P(X < 3)
- P( X > 4)
Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of X.
Choose the correct option from the given alternative:
Find expected value of and variance of X for the following p.m.f.
| X | -2 | -1 | 0 | 1 | 2 |
| P(x) | 0.3 | 0.3 | 0.1 | 0.05 | 0.25 |
Solve the following :
The following probability distribution of r.v. X
| X=x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| P(X=x) | 0.05 | 0.1 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |
Find the probability that
X is positive
Let X be amount of time for which a book is taken out of library by randomly selected student and suppose X has p.d.f
f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.
Calculate: P(x≤1)
Find the probability distribution of number of number of tails in three tosses of a coin
Find expected value and variance of X, the number on the uppermost face of a fair die.
70% of the members favour and 30% oppose a proposal in a meeting. The random variable X takes the value 0 if a member opposes the proposal and the value 1 if a member is in favour. Find E(X) and Var(X).
X is r.v. with p.d.f. f(x) = `"k"/sqrt(x)`, 0 < x < 4 = 0 otherwise then x E(X) = _______
Solve the following problem :
Find the expected value and variance of the r. v. X if its probability distribution is as follows.
| x | – 1 | 0 | 1 |
| P(X = x) | `(1)/(5)` | `(2)/(5)` | `(2)/(5)` |
Solve the following problem :
Find the expected value and variance of the r. v. X if its probability distribution is as follows.
| X | 0 | 1 | 2 | 3 | 4 | 5 |
| P(X = x) | `(1)/(32)` | `(5)/(32)` | `(10)/(32)` | `(10)/(32)` | `(5)/(32)` | `(1)/(32)` |
Solve the following problem :
Let X∼B(n,p) If E(X) = 5 and Var(X) = 2.5, find n and p.
If X denotes the number on the uppermost face of cubic die when it is tossed, then E(X) is ______
If a d.r.v. X takes values 0, 1, 2, 3, … with probability P(X = x) = k(x + 1) × 5–x, where k is a constant, then P(X = 0) = ______
If a d.r.v. X has the following probability distribution:
| X | –2 | –1 | 0 | 1 | 2 | 3 |
| P(X = x) | 0.1 | k | 0.2 | 2k | 0.3 | k |
then P(X = –1) is ______
Choose the correct alternative:
f(x) is c.d.f. of discete r.v. X whose distribution is
| xi | – 2 | – 1 | 0 | 1 | 2 |
| pi | 0.2 | 0.3 | 0.15 | 0.25 | 0.1 |
then F(– 3) = ______
Using the following activity, find the expected value and variance of the r.v.X if its probability distribution is as follows.
| x | 1 | 2 | 3 |
| P(X = x) | `1/5` | `2/5` | `2/5` |
Solution: µ = E(X) = `sum_("i" = 1)^3 x_"i""p"_"i"`
E(X) = `square + square + square = square`
Var(X) = `"E"("X"^2) - {"E"("X")}^2`
= `sum"X"_"i"^2"P"_"i" - [sum"X"_"i""P"_"i"]^2`
= `square - square`
= `square`
If F(x) is distribution function of discrete r.v.x with p.m.f. P(x) = `(x - 1)/(3)`; for x = 0, 1 2, 3, and P(x) = 0 otherwise then F(4) = _______.
The probability distribution of X is as follows:
| x | 0 | 1 | 2 | 3 | 4 |
| P[X = x] | 0.1 | k | 2k | 2k | k |
Find
- k
- P[X < 2]
- P[X ≥ 3]
- P[1 ≤ X < 4]
- P(2)
