Advertisements
Advertisements
Question
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 ≥ 1.5)
Advertisements
Solution
f(x) = 0.5 x , 0 ≤ x ≤ 2
= 0 , otherwise
P(x ≥ 1.5)
= `int_(1.5)^2 f(x) dx`
= `int_(1.5)^2 0.5x dx`
= ` 0.5[x^2/2]_(1.5)^2`
= `0.5[4/2 - 2.25/2]`
= `0.5 xx 1.75/2`
= `0.875/2`
= `0.875/2 xx 1000/1000`
= `(875 ÷ 125)/(2000 ÷ 125)`
= `7/16`
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) | 0.4 | 0.4 | 0.2 |
State if the following is not the probability mass function of a random variable. Give reasons for your answer
| Z | 3 | 2 | 1 | 0 | −1 |
| P(Z) | 0.3 | 0.2 | 0.4 | 0 | 0.05 |
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 |
Find k if the following function represent p.d.f. of r.v. X
f (x) = kx, for 0 < x < 2 and = 0 otherwise, Also find P `(1/ 4 < x < 3 /2)`.
Find k, if the following function represents p.d.f. of r.v. X.
f(x) = kx(1 – x), for 0 < x < 1 and = 0, otherwise.
Also, find `P(1/4 < x < 1/2) and P(x < 1/2)`.
Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by f(x) = `1/5`, for 0 ≤ x ≤ 5 and = 0 otherwise.
Find the probability that the waiting time is more than 4 minutes.
Choose the correct option from the given alternative:
If the p.d.f of a.c.r.v. X is f (x) = 3 (1 − 2x2 ), for 0 < x < 1 and = 0, otherwise (elsewhere) then the c.d.f of X is F(x) =
If the p.d.f. of c.r.v. X is f(x) = `x^2/18`, for -3 < x < 3 and = 0, otherwise, then P(|X| < 1) = ______.
Choose the correct option from the given alternative:
If p.m.f. of a d.r.v. X is P (X = x) = `((c_(x)^5 ))/2^5` , for x = 0, 1, 2, 3, 4, 5 and = 0, otherwise If a = P (X ≤ 2) and b = P (X ≥ 3), then E (X ) =
Choose the correct option from the given alternative:
If p.m.f. of a d.r.v. X is P (X = x) = `x^2 /(n (n + 1))`, for x = 1, 2, 3, . . ., n and = 0, otherwise then E (X ) =
Choose the correct option from the given alternative :
If p.m.f. of a d.r.v. X is P (x) = `c/ x^3` , for x = 1, 2, 3 and = 0, otherwise (elsewhere) then E (X ) =
Choose the correct option from the given alternative:
If the 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) =
Solve the following :
Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.
Amount of syrup prescribed by physician.
Solve the following :
Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.
The person on the high protein diet is interested gain of weight in a week.
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
The following is the c.d.f. of r.v. X
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| F(X) | 0.1 | 0.3 | 0.5 | 0.65 | 0.75 | 0.85 | 0.9 |
*1 |
P (–1 ≤ X ≤ 2)
The following is the c.d.f. of r.v. X:
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
| F(X) | 0.1 | 0.3 | 0.5 | 0.65 | 0.75 | 0.85 | 0.9 |
1 |
P (X ≤ 3/ X > 0)
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.
Given that X ~ B(n, p), if n = 10 and p = 0.4, find E(X) and Var(X)
Given that X ~ B(n, p), if n = 10, E(X) = 8, find Var(X).
Choose the correct alternative :
X: is number obtained on upper most face when a fair die….thrown then E(X) = _______.
The expected value of the sum of two numbers obtained when two fair dice are rolled is ______.
Fill in the blank :
E(x) is considered to be _______ of the probability distribution of x.
State whether the following is True or False :
If P(X = x) = `"k"[(4),(x)]` for x = 0, 1, 2, 3, 4 , then F(5) = `(1)/(4)` when F(x) is c.d.f.
State whether the following is True or False :
| x | – 2 | – 1 | 0 | 1 | 2 |
| P(X = x) | 0.2 | 0.3 | 0.15 | 0.25 | 0.1 |
If F(x) is c.d.f. of discrete r.v. X then F(–3) = 0
Solve the following problem :
The p.m.f. of a r.v.X is given by
`P(X = x) = {(((5),(x)) 1/2^5", ", x = 0", "1", "2", "3", "4", "5.),(0,"otherwise"):}`
Show that P(X ≤ 2) = P(X ≤ 3).
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)` |
The p.m.f. of a d.r.v. X is P(X = x) = `{{:(((5),(x))/2^5",", "for" x = 0"," 1"," 2"," 3"," 4"," 5),(0",", "otherwise"):}` If a = P(X ≤ 2) and b = P(X ≥ 3), then
If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(x/("n"("n" + 1))",", "for" x = 1"," 2"," 3"," .... "," "n"),(0",", "otherwise"):}`, then E(X) = ______
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 ______
If X is discrete random variable takes the values x1, x2, x3, … xn, then `sum_("i" = 1)^"n" "P"(x_"i")` = ______
The probability distribution of a discrete r.v. X is as follows:
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | k | 2k | 3k | 4k | 5k | 6k |
- Determine the value of k.
- Find P(X ≤ 4)
- P(2 < X < 4)
- P(X ≥ 3)
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)
Given below is the probability distribution of a discrete random variable x.
| X | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | K | 0 | 2K | 5K | K | 3K |
Find K and hence find P(2 ≤ x ≤ 3)
