Advertisements
Advertisements
Question
Verify which of the following is p.d.f. of r.v. X:
f(x) = 2, for 0 ≤ x ≤ 1.
Solution
f (x) is the p.d.f. of r.v. X if
(a) f (x) ≥ 0 for all x ∈ R and
(b) `int_(- ∞)^∞ f(x) dx = 1`
(a) f(x) = 2 ≥ 0 for 0 ≤ x ≤ 1
`int_(- ∞)^∞ f(x) dx = int_(- ∞)^0 f(x) dx + int_(0)^1 f(x) dx +int_( 1)^∞ f(x) dx`
= 0 +`int_0^1 2dx`+ 0
= [2x]01
= 2 - 0
= 2 ≠ 1
Hence, f (x) is not p.d.f. of X.
APPEARS IN
RELATED QUESTIONS
Verify which of the following is p.d.f. of r.v. X:
f(x) = sin x, for 0 ≤ x ≤ `π/2`
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
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) = 2 for 0 < x < q.
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(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(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 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)
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(1 < X < 2)
Choose the correct alternative :
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 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(–1 < X < 1)
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 :
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
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"):}`
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 ______.
