Advertisements
Advertisements
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.
Advertisements
उत्तर
Required probability = P(X > 33)
= `int_33^∞ f(x) dx`
= `int_33^35 f(x) dx + int_35^∞ f(x) dx`
= `int_33^35 f(x) dx + 0` ...[ f(x) = 0, when x > 35]
= `int_33^35 1/10 dx`
= `1/10 int_33^35 1dx`
= `1/10[x]_33^35`
= `1/10[35 - 33]`
= `2/10`
= `1/5`
Let F (x) be the c.d.f. of X
∴ F(x) = P[X ≤ x]
= `int_-∞^x f(x) dx`
= `int_-∞^25 f(x)dx + int_25^x f(x)dx`
= `0 + int_25^x f(x)dx` ...[∵ f(x) = 0, when f(x) < 25]
= `int_25^x 1/10dx`
= `1/10 int_25^x 1dx`
= `1/10[x]_25^x`
= `1/10[x - 25]`
∴ F(x) = `(x - 25)/10`
APPEARS IN
संबंधित प्रश्न
Verify which of the following is p.d.f. of r.v. X:
f(x) = sin x, for 0 ≤ x ≤ `π/2`
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
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),
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 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(–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 < –2)
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 > 0)
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)
If a r.v. X has p.d.f f(x) = `{("c"/x"," 1 < x < 3"," "c" > 0),(0"," "otherwise"):}`
Find c, E(X), and Var(X). Also Find F(x).
Given p.d.f. of a continuous r.v.X as
f(x) = `x^2/(3)` for –1 < x < 2
= 0, otherwise
then F(1) = _______.
State whether the following is True or False :
If f(x) = k x (1 – x) for 0 < x < 1 = 0 otherwise k = 12
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(X < – 0.5 or X > 0.5)
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).
The values of continuous r.v. are generally obtained by ______
If r.v. X assumes the values 1, 2, 3, …….., 9 with equal probabilities, then E(X) = 5
State whether the following statement is True or False:
The cumulative distribution function (c.d.f.) of a continuous random variable X is denoted by F and defined by
F(x) = `{:(0",", "for all" x ≤ "a"),( int_"a"^x f(x) "d"x",", "for all" x ≥ "a"):}`
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.
