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प्रश्न
A random variable X has the following probability distribution
| X | 2 | 3 | 4 |
| P(x) | 0.3 | 0.4 | 0.3 |
Then the variance of this distribution is
विकल्प
0.6
0.7
0.77
0.66
उत्तर
0.6
संबंधित प्रश्न
From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.
Probability distribution of X is given by
| X = x | 1 | 2 | 3 | 4 |
| P(X = x) | 0.1 | 0.3 | 0.4 | 0.2 |
Find P(X ≥ 2) and obtain cumulative distribution function of X
State the following are not the probability distributions of a random variable. Give reasons for your answer.
| X | 0 | 1 | 2 |
| P (X) | 0.4 | 0.4 | 0.2 |
State the following are not the probability distributions of a random variable. Give reasons for your answer.
| Z | 3 | 2 | 1 | 0 | -1 |
| P(Z) | 0.3 | 0.2 | 0.4 | 0.1 | 0.05 |
Find the probability distribution of number of heads in four tosses of a coin.
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.
An urn contains 25 balls of which 10 balls bear a mark ‘X’ and the remaining 15 bear a mark ‘Y’. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that
(i) all will bear ‘X’ mark.
(ii) not more than 2 will bear ‘Y’ mark.
(iii) at least one ball will bear ‘Y’ mark
(iv) the number of balls with ‘X’ mark and ‘Y’ mark will be equal.
A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).
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.
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.
Which of the following distributions of probabilities of a random variable X are the probability distributions?
(i)
| X : | 3 | 2 | 1 | 0 | −1 |
| P (X) : | 0.3 | 0.2 | 0.4 | 0.1 | 0.05 |
| X : | 0 | 1 | 2 |
| P (X) : | 0.6 | 0.4 | 0.2 |
(iii)
| X : | 0 | 1 | 2 | 3 | 4 |
| P (X) : | 0.1 | 0.5 | 0.2 | 0.1 | 0.1 |
(iv)
| X : | 0 | 1 | 2 | 3 |
| P (X) : | 0.3 | 0.2 | 0.4 | 0.1 |
A random variable X has the following probability distribution:
| Values of X : | −2 | −1 | 0 | 1 | 2 | 3 |
| P (X) : | 0.1 | k | 0.2 | 2k | 0.3 | k |
Find the value of k.
A random variable X has the following probability distribution:
| Values of X : | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P (X) : | a | 3a | 5a | 7a | 9a | 11a | 13a | 15a | 17a |
Determine:
(i) The value of a
(ii) P (X < 3), P (X ≥ 3), P (0 < X < 5).
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: c
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)
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 (1 < X ≤ 2)
A random variable X takes the values 0, 1, 2 and 3 such that:
P (X = 0) = P (X > 0) = P (X < 0); P (X = −3) = P (X = −2) = P (X = −1); P (X = 1) = P (X = 2) = P (X = 3) . Obtain the probability distribution of X.
A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance of being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X?
Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of kings.
Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls
An urn contains 4 red and 3 blue balls. Find the probability distribution of the number of blue balls in a random draw of 3 balls with replacement.
Let X represent the difference between the number of heads and the number of tails when a coin is tossed 6 times. What are the possible values of X?
From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of the number of defective bulbs.
Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red balls drawn, then find 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 distributions:
| xi : | 2 | 3 | 4 |
| pi : | 0.2 | 0.5 | 0.3 |
Find the mean and standard deviation of each of the following probability distribution:
| xi : | 1 | 3 | 4 | 5 |
| pi: | 0.4 | 0.1 | 0.2 | 0.3 |
Find the mean and standard deviation of each of the following probability distribution :
| xi : | -5 | -4 | 1 | 2 |
| pi : | \[\frac{1}{4}\] | \[\frac{1}{8}\] | \[\frac{1}{2}\] | \[\frac{1}{8}\] |
Find the mean and standard deviation of each of the following probability distribution :
| xi: | 0 | 1 | 3 | 5 |
| pi : | 0.2 | 0.5 | 0.2 | 0.1 |
Find the mean and standard deviation of each of the following probability distribution :
| xi : | -2 | -1 | 0 | 1 | 2 |
| pi : | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |
Find the mean and standard deviation of each of the following probability distribution :
| xi : | -3 | -1 | 0 | 1 | 3 |
| pi : | 0.05 | 0.45 | 0.20 | 0.25 | 0.05 |
A discrete random variable X has the probability distribution given below:
| X: | 0.5 | 1 | 1.5 | 2 |
| P(X): | k | k2 | 2k2 | k |
Find the value of k.
Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
A box contains 13 bulbs, out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement, from the box. Find the probability distribution of the number of defective bulbs.
For what value of k the following distribution is a probability distribution?
| X = xi : | 0 | 1 | 2 | 3 |
| P (X = xi) : | 2k4 | 3k2 − 5k3 | 2k − 3k2 | 3k − 1 |
If the probability distribution of a random variable X is as given below:
Write the value of P (X ≤ 2).
| X = xi : | 1 | 2 | 3 | 4 |
| P (X = xi) : | c | 2c | 4c | 4c |
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
A random variable X takes the values 0, 1, 2, 3 and its mean is 1.3. If P (X = 3) = 2 P (X = 1) and P (X = 2) = 0.3, then P (X = 0) is
Mark the correct alternative in the following question:
The probability distribution of a discrete random variable X is given below:
| X: | 2 | 3 | 4 | 5 |
| P(X): |
\[\frac{5}{k}\]
|
\[\frac{7}{k}\]
|
\[\frac{9}{k}\]
|
\[\frac{11}{k}\] |
The value of k is .
Mark the correct alternative in the following question:
For the following probability distribution:
| X : | 1 | 2 | 3 | 4 |
| P(X) : |
\[\frac{1}{10}\]
|
\[\frac{1}{5}\]
|
\[\frac{3}{10}\]
|
\[\frac{2}{5}\]
|
The value of E(X2) 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.
Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.
Calculate `"e"_0^circ ,"e"_1^circ , "e"_2^circ` from the following:
| Age x | 0 | 1 | 2 |
| lx | 1000 | 880 | 876 |
| Tx | - | - | 3323 |
Demand function x, for a certain commodity is given as x = 200 - 4p where p is the unit price. Find :
(a) elasticity of demand as function of p.
(b) elasticity of demand when p = 10 , interpret your result.
If the demand function is D = 150 - p2 - 3p, find marginal revenue, average revenue and elasticity of demand for price p = 3.
Verify the following function, which can be regarded as p.m.f. for the given values of X :
| X = x | -1 | 0 | 1 |
| P(x) | -0.2 | 1 | 0.2 |
Compute the age specific death rate for the following data :
| Age group (years) | Population (in thousands) | Number of deaths |
| Below 5 | 15 | 360 |
| 5-30 | 20 | 400 |
| Above 30 | 10 | 280 |
A departmental store gives trafnfng to the salesmen in service followed by a test. It is experienced that the performance regarding sales of any salesman is linearly related to the scores secured by him. The following data gives the test scores and sales made by nine (9) salesmen during a fixed period.
| Test scores (X) | 16 | 22 | 28 | 24 | 29 | 25 | 16 | 23 | 24 |
| Sales (Y) (₹ in hundreds) | 35 | 42 | 57 | 40 | 54 | 51 | 34 | 47 | 45 |
(a) Obtain the line of regression of Y on X.
(b) Estimate Y when X = 17.
A fair coin is tossed 12 times. Find the probability of getting at least 2 heads .
If X ∼ N (4,25), then find P(x ≤ 4)
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]
Amit and Rohit started a business by investing ₹20,000 each. After 3 months Amit withdrew ₹5,000 and Rohit put in ₹5,000 additionally. How should a profit of ₹12,800 be divided between them at the end of the year?
A random variable X has the following probability distribution :
| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X) | C | 2C | 2C | 3C | C2 | 2C2 | 7C2+C |
Find the value of C and also calculate the mean of this distribution.
Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.
A random variable X has the following probability distribution :
| x = x | 0 | 1 | 2 | 3 | 7 | |||
| P(X=x) | 0 | k | 2k | 2k | 3k | k2 | 2k2 | 7k2 + k |
Determine (i) k
(ii) P(X> 6)
(iii) P(0<X<3).
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 |
A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. If X denotes the age of a randomly selected student, find the probability distribution of X. Find the mean and variance of X.
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 successes
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at most 2 successes.
There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item?
State whether the following is True or False :
If r.v. X assumes the values 1, 2, 3, ……. 9 with equal probabilities, E(x) = 5.
Solve the following problem :
Following is the probability distribution of a r.v.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 even.
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 :
Find the probability of the number of successes in two tosses of a die, where success is defined as six appears in at least one toss.
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 :
A computer installation has 3 terminals. The probability that any one terminal requires attention during a week is 0.1, independent of other terminals. Find the probabilities that 1 terminal requires attention during a week.
Solve the following problem :
In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.
Calculate the probabilities of obtaining an answer yes from all of the selected students.
Solve the following problem :
In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.
Find the probability that the visitor obtains the answer yes from at least 3 students.
Solve the following problem :
It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on at most 2 days of a week.
Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red ball drawn, find the probability distribution of X.
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 |
Find the value of k
Consider the probability distribution of a random variable X:
| X | 0 | 1 | 2 | 3 | 4 |
| P(X) | 0.1 | 0.25 | 0.3 | 0.2 | 0.15 |
Calculate `"V"("X"/2)`
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.
The random variable X can take only the values 0, 1, 2. Given that P(X = 0) = P(X = 1) = p and that E(X2) = E[X], find the value of p
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 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)
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 Standard deviation of X.
The probability distribution of a random variable x is given as under:
P(X = x) = `{{:("k"x^2, "for" x = 1"," 2"," 3),(2"k"x, "for" x = 4"," 5"," 6),(0, "otherwise"):}`
where k is a constant. Calculate P(X ≥ 4)
For the following probability distribution:
| X | – 4 | – 3 | – 2 | – 1 | 0 |
| P(X) | 0.1 | 0.2 | 0.3 | 0.2 | 0.2 |
E(X) is equal to ______.
For the following probability distribution:
| X | 1 | 2 | 3 | 4 |
| P(X) | `1/10` | `3/10` | `3/10` | `2/5` |
E(X2) is equal to ______.
A bag contains 1 red and 3 white balls. Find the probability distribution of the number of red balls if 2 balls are drawn at random from the bag one-by-one without replacement.
Two numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of two numbers selected, find the probability distribution 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`
