Advertisements
Advertisements
प्रश्न
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 k, if f(x) = `{:(kx^2(1 - x)",", "for" 0 < x < 1),(0",", "otherwise"):}`
is the p.d.f. of random variable X.
उत्तर १
Since f(x) is p.d.f. of r.v.X, we get
`int_-∞^∞ f(x) dx` = 1
∴ `int_-∞^0 f(x)dx + int_0^1 f(x)dx + int_1^∞ f(x)dx` = 1
∴ `0 + int_0^1 kx^2 (1 - x)dx + 0` = 1
∴ `kint_0^1 (x^2 - x^3)dx` = 1
∴ `k[x^3/3 - x^4/4]_0^1` = 1
∴ `k[(1/3 - 1/4) - 0]` = 1
∴ `k(1/12)` = 1
∴ k = 12
उत्तर २
Since (x) is the p.d.f. of a r.v. X,
∴ `int_0^1 kx^2 (1 - x) dx = 1`
∴ `int_0^1 k (x^2 - x^3)dx = 1`
`k int_0^1 (x^2 - x^3) dx = 1`
∴ `k{int_0^1 x^2 dx - int_0^1 x^3 dx} = 1`
∴ `k [x^3/3 - x^4/4]_0^1 = 1`
∴ `k(1/12) = 1`
∴ k = 12
APPEARS IN
संबंधित प्रश्न
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) = x, for 0 ≤ x ≤ 1 and 2 - x for 1 < x < 2
Verify which of the following is p.d.f. of r.v. X:
f(x) = 2, for 0 ≤ x ≤ 1.
It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by
f (x) = `x^2/ 3` , for –1 < x < 2 and = 0 otherwise
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 | 0.10 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |
Find the probability that
X is non-negative
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 | 0.10 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |
Find the probability that
X is odd
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 | 0.10 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |
Find the probability that
X is even
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 < 1.5),
The following is the p.d.f. of a r.v. X.
f(x) = `{(x/(8), "for" 0 < x < 4),(0, "otherwise."):}`
Find P(1 < 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(0.5 ≤ X ≤ 1.5)
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 X is the waiting time (in minutes) for a bus and its p. d. f. is given by
f(x) = `{(1/5, "for" 0 ≤ x ≤ 5),(0, "otherwise".):}`
Find the probability that waiting time is more than 4 minutes.
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)
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)
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)
State whether the following is True or False :
If f(x) = k x (1 – x) for 0 < x < 1 = 0 otherwise k = 12
State whether the following is True or False :
If X ~ B(n,p) and n = 6 and P(X = 4) = P(X = 2) then p = `(1)/(2)`
Solve the following problem :
In the following probability distribution of a r.v.X.
| x | 1 | 2 | 3 | 4 | 5 |
| P (x) | `(1)/(20)` | `(3)/(20)` | a | 2a | `(1)/(20)` |
Find a and obtain the c.d.f. of 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(X > 0)
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)
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(X < – 0.5 or X > 0.5)
Solve the following problem :
The p.d.f. of the r.v. X is given by
f(x) = `{((1)/(2"a")",", "for" 0 < x= 2"a".),(0, "otherwise".):}`
Show that `"P"("X" < "a"/2) = "P"("X" > (3"a")/2)`
Solve the following problem :
Determine k if the p.d.f. of the r.v. is
f(x) = `{("ke"^(-thetax), "for" 0 ≤ x < oo),(0, "otherwise".):}`
Find `"P"("X" > 1/theta)` and determine M is P(0 < X < M) = `(1)/(2)`
Solve the following problem :
The p.d.f. of the r.v. X is given by
f(x) = `{("k"/sqrt(x), "for" 0 < x < 4.),(0, "otherwise".):}`
Determine k, the c.d.f. of X, and hence find P(X ≤ 2) and P(X ≥ 1).
Solve the following problem :
Let X denote the reaction temperature in Celsius of a certain chemical process. Let X have the p. d. f.
f(x) = `{((1)/(10), "for" -5 ≤ x < 5),(0, "otherwise".):}`
Compute P(X < 0).
The values of continuous r.v. are generally obtained by ______
State whether the following statement is True or False:
If f(x) = `{:("k"x (1 - x)",", "for" 0 < x < 1),(= 0",", "otherwise"):}`
is the p.d.f. of a r.v. X, then k = 12
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"):}`
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"):}`
If the p.d.f. of X is
f(x) = `x^2/18, - 3 < x < 3`
= 0, otherwise
Then P(X < 1) is ______.
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.
