Advertisements
Advertisements
प्रश्न
If random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)
उत्तर १
∴ X has probability distribution function
∴ `int_1^3 c/x dx = 1`
`[ c log x]_1^3` = 1
c [ log3 - log 1 ] = 1
c log 3 = 1
c = `1/log 3`
E(X) = `int_1^3 xf(x)`
= `int_1^3 x . c/x dx`
= `c int_1^3 1 dx`
= `c | x |_1^3`
= c | 3 - 1|
= 2c
E(X) = `2/log 3`
V(X) = `E( "X"^2 ) - E( "X"^2 )`
V(X) = `int_1^3 x^2f(x) dx - [ int_1^3 x x f(x) dx ]^2`
= `int_1^3 x^2 c/x dx - [ 2/log 3 ]^2`
= `int_1^3 cx dx - [ 2/log 3 ]^2`
= `c[ x^2/2 ]_1^3 - [ 2/log 3 ]^2`
= `1/log 3 xx [ (9 - 1)/2] - 4/[(log 3)^2]`
V(X) = = `4/log 3- 4/[(log 3)^2]`
उत्तर २
∴ X has probability distribution function
∴ `int_1^3 c/x dx = 1`
`[ c log x]_1^3` = 1
c [ log3 - log 1 ] = 1
c log 3 = 1
c = `1/log 3`
E(X) = `int_1^3 xf(x)`
= `int_1^3 x . c/x dx`
= `c int_1^3 1 dx`
= `c | x |_1^3`
= c | 3 - 1|
= 2c
E(X) = `2/log 3`
V(X) = `E( "X"^2 ) - E( "X"^2 )`
V(X) = `int_1^3 x^2f(x) dx - [ int_1^3 x x f(x) dx ]^2`
= `int_1^3 x^2 c/x dx - [ 2/log 3 ]^2`
= `int_1^3 cx dx - [ 2/log 3 ]^2`
= `c[ x^2/2 ]_1^3 - [ 2/log 3 ]^2`
= `1/log 3 xx [ (9 - 1)/2] - 4/[(log 3)^2]`
V(X) = = `4/log 3- 4/[(log 3)^2]`
उत्तर ३
∴ X has probability distribution function
∴ `int_1^3 c/x dx = 1`
`[ c log x]_1^3` = 1
c [ log3 - log 1 ] = 1
c log 3 = 1
c = `1/log 3`
E(X) = `int_1^3 xf(x)`
= `int_1^3 x . c/x dx`
= `c int_1^3 1 dx`
= `c | x |_1^3`
= c | 3 - 1|
= 2c
E(X) = `2/log 3`
V(X) = `E( "X"^2 ) - E( "X"^2 )`
V(X) = `int_1^3 x^2f(x) dx - [ int_1^3 x x f(x) dx ]^2`
= `int_1^3 x^2 c/x dx - [ 2/log 3 ]^2`
= `int_1^3 cx dx - [ 2/log 3 ]^2`
= `c[ x^2/2 ]_1^3 - [ 2/log 3 ]^2`
= `1/log 3 xx [ (9 - 1)/2] - 4/[(log 3)^2]`
V(X) = = `4/log 3- 4/[(log 3)^2]`
उत्तर ४
∴ X has probability distribution function
∴ `int_1^3 c/x dx = 1`
`[ c log x]_1^3` = 1
c [ log3 - log 1 ] = 1
c log 3 = 1
c = `1/log 3`
E(X) = `int_1^3 xf(x)`
= `int_1^3 x . c/x dx`
= `c int_1^3 1 dx`
= `c | x |_1^3`
= c | 3 - 1|
= 2c
E(X) = `2/log 3`
V(X) = `E( "X"^2 ) - E( "X"^2 )`
V(X) = `int_1^3 x^2f(x) dx - [ int_1^3 x x f(x) dx ]^2`
= `int_1^3 x^2 c/x dx - [ 2/log 3 ]^2`
= `int_1^3 cx dx - [ 2/log 3 ]^2`
= `c[ x^2/2 ]_1^3 - [ 2/log 3 ]^2`
= `1/log 3 xx [ (9 - 1)/2] - 4/[(log 3)^2]`
V(X) = = `4/log 3- 4/[(log 3)^2]`
APPEARS IN
संबंधित प्रश्न
State the following are not the probability distributions of a random variable. Give reasons for your answer.
| Y | -1 | 0 | 1 |
| P(Y) | 0.6 | 0.1 | 0.2 |
From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Assume that the chances of the patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean 'and variance of X.
Three persons A, B and C shoot to hit a target. If A hits the target four times in five trials, B hits it three times in four trials and C hits it two times in three trials, find the probability that:
1) Exactly two persons hit the target.
2) At least two persons hit the target.
3) None hit the target.
The probability distribution function of a random variable X is given by
| xi : | 0 | 1 | 2 |
| pi : | 3c3 | 4c − 10c2 | 5c-1 |
where c > 0 Find: P (X < 2)
Find the probability distribution of the number of heads, when three coins are tossed.
A bag contains 4 red and 6 black balls. Three balls are drawn at random. Find the probability distribution of the number of red balls.
Two dice are thrown together and the number appearing on them noted. X denotes the sum of the two numbers. Assuming that all the 36 outcomes are equally likely, what is the probability distribution of X?
The probability distribution of a random variable X is given below:
| x | 0 | 1 | 2 | 3 |
| P(X) | k |
\[\frac{k}{2}\]
|
\[\frac{k}{4}\]
|
\[\frac{k}{8}\]
|
Find P(X ≤ 2) + P(X > 2) .
Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.
Find the mean and standard deviation of each of the following probability distribution:
| xi : | −1 | 0 | 1 | 2 | 3 |
| pi : | 0.3 | 0.1 | 0.1 | 0.3 | 0.2 |
If a random variable X has the following probability distribution:
| X : | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P (X) : | a | 3a | 5a | 7a | 9a | 11a | 13a | 15a | 17a |
then the value of a is
A random variable X has the following probability distribution:
| X : | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P (X) : | 0.15 | 0.23 | 0.12 | 0.10 | 0.20 | 0.08 | 0.07 | 0.05 |
For the events E = {X : X is a prime number}, F = {X : X < 4}, the probability P (E ∪ F) is
An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.
Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).
Three different aeroplanes are to be assigned to carry three cargo consignments with a view to maximize profit. The profit matrix (in lakhs of ₹) is as follows :
| Aeroplanes | Cargo consignments | ||
| C1 | C2 | C3 | |
| A1 | 1 | 4 | 5 |
| A2 | 2 | 3 | 3 |
| A3 | 3 | 1 | 2 |
How should the cargo consignments be assigned to the aeroplanes to maximize the profit?
The expenditure Ec of a person with income I is given by Ec = (0.000035) I2 + (0. 045) I. Find marginal propensity to consume (MPC) and average propensity to consume (APC) when I = 5000.
The defects on a plywood sheet occur at random with an average of the defect per 50 sq. ft. What Is the probability that such sheet will have-
(a) No defects
(b) At least one defect
[Use e-1 = 0.3678]
Solve the following:
Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.
A highway safety group is interested in studying the speed (km/hrs) of a car at a check point.
Determine whether each of the following is a probability distribution. Give reasons for your answer.
| x | 0 | 1 | 2 |
| P(x) | 0.4 | 0.4 | 0.2 |
Determine whether each of the following is a probability distribution. Give reasons for your answer.
| z | 3 | 2 | 1 | 0 | -1 |
| P(z) | 0.3 | 0.2 | 0.4. | 0.05 | 0.05 |
Determine whether each of the following is a probability distribution. Give reasons for your answer.
| x | 0 | 1 | 2 |
| P(x) | 0.3 | 0.4 | 0.2 |
Defects on plywood sheet occur at random with the average of one defect per 50 sq.ft. Find the probability that such a sheet has:
- no defect
- at least one defect
Use e−1 = 0.3678
Solve the following problem :
Find the probability of the number of successes in two tosses of a die, where success is defined as number greater than 4.
Solve the following problem :
If a fair coin is tossed 4 times, find the probability that it shows 3 heads
Solve the following problem :
If a fair coin is tossed 4 times, find the probability that it shows head in the first 2 tosses and tail in last 2 tosses.
Solve the following problem :
A large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Find the probability that the inspector finds at most one defective item in the 4 selected items.
Solve the following problem :
The probability that a machine will produce all bolts in a production run within the specification is 0.9. A sample of 3 machines is taken at random. Calculate the probability that all machines will produce all bolts in a production run within the specification.
Solve the following problem :
It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on exactly 3 days of a week.
For the random variable X, if V(X) = 4, E(X) = 3, then E(x2) is ______
A discrete random variable X has the probability distribution given as below:
| X | 0.5 | 1 | 1.5 | 2 |
| P(X) | k | k2 | 2k2 | k |
Determine the mean of the distribution.
The probability distribution of a random variable X is given below:
| X | 0 | 1 | 2 | 3 |
| P(X) | k | `"k"/2` | `"k"/4` | `"k"/8` |
Determine the value of k.
Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1), "for" x = 1"," 2"," 3"," 4),(2"k"x, "for" x = 5"," 6"," 7),(0, "Otherwise"):}`
where k is a constant. Calculate E(X)
The probability distribution of a discrete random variable X is given as under:
| X | 1 | 2 | 4 | 2A | 3A | 5A |
| P(X) | `1/2` | `1/5` | `3/25` | `1/10` | `1/25` | `1/25` |
Calculate: Variance of X
The probability that a bomb will hit the target is 0.8. Complete the following activity to find, the probability that, out of 5 bombs exactly 2 will miss the target.
Solution: Here, n = 5, X =number of bombs that hit the target
p = probability that bomb will hit the target = `square`
∴ q = 1 - p = `square`
Here, `X∼B(5,4/5)`
∴ P(X = x) = `""^"n""C"_x"P"^x"q"^("n" - x) = square`
P[Exactly 2 bombs will miss the target] = P[Exactly 3 bombs will hit the target]
= P(X = 3)
=`""^5"C"_3(4/5)^3(1/5)^2=10(4/5)^3(1/5)^2`
∴ P(X = 3) = `square`
A large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Complete the following activity to find the probability that the inspector finds at most one defective item in the 4 selected items.
Solution:
Here, n = 4
p = probability of defective device = 10% = `10/100 = square`
∴ q = 1 - p = 1 - 0.1 = `square`
X ∼ B(4, 0.1)
`P(X=x)=""^n"C"_x p^x q^(n-x)= ""^4"C"_x (0.1)^x (0.9)^(4 - x)`
P[At most one defective device] = P[X ≤ 1]
= P[X=0] + P[X=1]
= `square+square`
∴ P[X ≤ 1] = `square`
