Advertisements
Advertisements
Question
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"):}`
Solution
P(X < 1) = `int_-2^1 (x + 2)/18 dx`
= `1/18 [x^2/2 + 2x]_-2^1`
= `1/18 {(1/2 + 2) - ((-2)^2/2 + 2(-2))}`
= `1/18 {5/2 + 2}`
= `1/18 xx 9/2`
= `1/4`
P(X < 1) = `1/4`
APPEARS IN
RELATED QUESTIONS
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.):}`
Check whether the following is a p.d.f.
f(x) = 2 for 0 < x < q.
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 between 1 and 3 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)
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(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)
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).
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) = _______.
Fill in the blank :
If x is continuous r.v. and F(xi) = P(X ≤ xi) = `int_(-oo)^(oo) f(x)*dx` then F(x) is called _______
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 :
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 ______
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"):}`
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"):}`
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.
