Advertisements
Advertisements
Question
Given the p.d.f. of a continuous r.v. X ,
f (x) = `x^2/3` , for –1 < x < 2 and = 0 otherwise
Determine c.d.f. of X hence find P(1 < x < 2)
Advertisements
Solution
Then F(x) = ` int_(-∞)^x f (x) dx`
=` int_(-∞)^-1 f (x) dx + int_(-1)^x f (x) dx`
= 0 + `int_(-1)^x x^2/3 dx = 1/3int_(-1)^x x^2 dx`
= `1/3[x^3/3]_-1^x`
= `1/3[x^3/3-(-1/3)]`
∴ f(x) = `(x^3+1)/9`
P (1 < x < 2) = F (2) - F (1)
= `((2^3+1)/9)- ( (1^3+1)/9)= 1-2/9=7/9`
APPEARS IN
RELATED QUESTIONS
Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f.
`"f(x)" = {("k"(4 - x^2) "for –2 ≤ x ≤ 2,"),(0 "otherwise".):}`
P(–1 < x < 1)
Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f.
f (x) = k `(4 – x^2)`, for –2 ≤ x ≤ 2 and = 0 otherwise.
P (–0·5 < x or x > 0·5)
The following is the p.d.f. of continuous r.v.
f (x) = `x/8` , for 0 < x < 4 and = 0 otherwise.
Find F(x) at x = 0·5 , 1.7 and 5
Given the p.d.f. of a continuous r.v. X ,
f (x) = `x^2/ 3` , for –1 < x < 2 and = 0 otherwise
Determine c.d.f. of X hence find P( X > 0)
Choose the correct option from the given alternative:
If the a d.r.v. X has the following probability distribution:
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P(X=x) | k | 2k | 2k | 3k | k2 | 2k2 | 7k2+k |
k =
In the p.m.f. of r.v. X
| X | 1 | 2 | 3 | 4 | 5 |
| P (X) | `1/20` | `3/20` | a | 2a | `1/20` |
Find a and obtain c.d.f. of X.
Solve the following problem :
A player tosses two coins. He wins ₹ 10 if 2 heads appear, ₹ 5 if 1 head appears, and ₹ 2 if no head appears. Find the expected value and variance of winning amount.
It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.
f(x) = `{(x^3/(64), "for" 0 ≤ x ≤ 4),(0, "otherwise."):}`
Verify whether f(x) is a p.d.f.
It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.
f(x) = `{(x^3/(64), "for" 0 ≤ x ≤ 4),(0, "otherwise."):}`
Find probability that X is between 1 and 3..
Solve the following problem :
Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
A person on high protein diet is interested in the weight gained in a week.
c.d.f. of a discrete random variable X is
The probability distribution of a r.v. X is
| X = x | -3 | -2 | -1 | 0 | 1 |
| P(X = x) | 0.3 | 0.2 | 0.25 | 0.1 | 0.15 |
Then F (-1) = ?
Out of 100 people selected at random, 10 have common cold. If five persons selected at random from the group, then the probability that at most one person will have common cold is ______.
A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find P(X ≥ 6)
Find the probability mass function and cumulative distribution function of a number of girl children in families with 4 children, assuming equal probabilities for boys and girls
Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by
`f(x) = {{:((x^2 + 1)/k"," "for" x = 0"," 1"," 2),(0"," "otherwise"):}`
Find the value of k
Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by
`f(x) = {{:((x^2 + 1)/k"," "for" x = 0"," 1"," 2),(0"," "otherwise"):}`
Find cumulative distribution function
The cumulative distribution function of a discrete random variable is given by
F(x) = `{{:(0, - oo < x < - 1),(0.15, - 1 ≤ x < 0),(0.35, 0 ≤ x < 1),(0.60, 1 ≤ x < 2),(0.85, 2 ≤ x < 3),(1, 3 ≤ x < oo):}`
Find the probability mass function
The cumulative distribution function of a discrete random variable is given by
F(x) = `{{:(0, - oo < x < - 1),(0.15, - 1 ≤ x < 0),(0.35, 0 ≤ x < 1),(0.60, 1 ≤ x < 2),(0.85, 2 ≤ x < 3),(1, 3 ≤ x < oo):}`
Find P(X ≥ 2)
A random variable X has the following probability mass function.
| x | 1 | 2 | 3 | 4 | 5 |
| F(x) | k2 | 2k2 | 3k2 | 2k | 3k |
Find P(2 ≤ X < 5)
The cumulative distribution function of a discrete random variable is given by
F(x) = `{{:(0, "for" - oo < x < 0),(1/2, "for" 0 ≤ x < 1),(3/5, "for" 1 ≤ x < 2),(4/5, "for" 2 ≤ x < 4),(9/5, "for" 3 ≤ x < 4),(1, "for" ≤ x < oo):}`
Find P(X < 3)
If Xis a.r.v. with c.d.f F (x) and its probability distribution is given by
| X = x | - 1.5 | -0.5 | 0.5 | 1.5 | 2.5 |
| P(X = x) | 0.05 | 0.2 | 0.15 | 0.25 | 0.35 |
then, F(1.5) - F(- 0.5) = ?
Choose the correct alternative:
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
Choose the correct alternative:
Which of the following is a discrete random variable?
I. The number of cars crossing a particular signal in a day.
II. The number of customers in a queue to buy train tickets at a moment.
III. The time taken to complete a telephone call.
The p.m.f. of a random variable X is
P(x) = `(5 - x)/10`, x = 1, 2, 3, 4
= 0, otherwise
The value of E(X) is ______
A bag contains 6 white and 4 black balls. Two balls are drawn at random. The probability that they are of the same colour is ______.
If the c.d.f (cumulative distribution function) is given by F(x) = `(x - 25)/10`, then P(27 ≤ x ≤ 33) = ______.
If the probability function of a random variable X is defined by P(X = k) = a`((k + 1)/2^k)` for k - 0, 1, 2, 3, 4, 5, then the probability that X takes a prime value is ______
For a random variable X, if Var (X) = 5 and E (X2) = 21, the value of E (X) is ______
X is a continuous random variable with a probability density function
f(x) = `{{:(x^2/4 + k; 0 ≤ x ≤ 2),(0; "otherwise"):}`
The value of k is equal to ______
The probability distribution of a random variable X is given below. If its mean is 4.2, then the values of a and bar respectively
| X = x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | a | a | a | b | b | 0.3 |
A card is chosen from a well-shuffled pack of cards. The probability of getting an ace of spade or a jack of diamond is ______.
The c.d.f. of a discrete r.v. x is
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| F(x) | 0.16 | 0.41 | 0.56 | 0.70 | 0.91 | 1.00 |
Then P(1 < x ≤ 4) = ______
The p.d.f. of a continuous random variable X is
f(x) = 0.1 x, 0 < x < 5
= 0, otherwise
Then the value of P(X > 3) is ______
A random variable X has the following probability distribution:
| X = xi | 1 | 2 | 3 | 4 |
| P(X = xi) | 0.2 | 0.15 | 0.3 | 0.35 |
The mean and the variance are respectively ______.
Two cards are randomly drawn, with replacement. from a well shuffled deck of 52 playing cards. Find the probability distribution of the number of aces drawn.
At random variable X – B(n, p), if values of mean and variance of X are 18 and 12 respectively, then total number of possible values of X are ______.
For the following distribution function F(x) of a rv.x.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| F(x) | 0.2 | 0.37 | 0.48 | 0.62 | 0.85 | 1 |
P(3 < x < 5) =
