Advertisements
Advertisements
प्रश्न
Find the mean of number randomly selected from 1 to 15.
उत्तर
The sample space of the experiment is S = {1, 2, 3, …, 15}.
Let X denotes the number selected.
Then X is a random variable that can take values 1, 2, 3, …, 15.
Each number selected is equiprobable therefore
P(1) = P(2) = P(3) = … = P(15) = `1/15`
μ = E(X) = `sum_(i = 1)^n x_ip_i`
= `1 xx 1/15 + 2 xx 1/15 + 3 xx 1/15 + ... + 15 xx 1/15`
= `(1 + 2 + 3 + ... + 15) xx 1/15`
= `((15 xx 16)/2) xx 1/15`
= 8
APPEARS IN
संबंधित प्रश्न
From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain -
(a) exactly 1 defective bulb.
(b) at least 1 defective bulb.
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 | 3 | 4 |
| P(X) | 0.1 | 0.5 | 0.2 | -0.1 | 0.3 |
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 |
Find the probability distribution of number of heads in two tosses of a coin.
Find the probability distribution of number of tails in the simultaneous tosses of three coins.
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 random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).
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.
Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution.
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.
Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Find the probability distribution of the number of heads, when three coins are tossed.
Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.
Five defective bolts are accidently mixed with twenty good ones. If four bolts are drawn at random from this lot, find the probability distribution of the number of defective bolts.
Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Two cards are drawn simultaneously from a well-shuffled deck of 52 cards. Find the probability distribution of the number of successes, when getting a spade is considered a success.
A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of number of successes.
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?
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 : | -3 | -1 | 0 | 1 | 3 |
| pi : | 0.05 | 0.45 | 0.20 | 0.25 | 0.05 |
Find the mean and standard deviation of each of the following probability distribution :
| xi : | 0 | 1 | 2 | 3 | 4 | 5 |
| pi : |
\[\frac{1}{6}\]
|
\[\frac{5}{18}\]
|
\[\frac{2}{9}\]
|
\[\frac{1}{6}\]
|
\[\frac{1}{9}\]
|
\[\frac{1}{18}\]
|
Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.
A fair coin is tossed four times. Let X denote the number of heads occurring. Find the probability distribution, mean and variance of X.
A fair die is tossed. Let X denote twice the number appearing. Find the probability distribution, mean and variance of X.
In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.
If X denotes the number on the upper face of a cubical die when it is thrown, find the mean of X.
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 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
A random variable has the following probability distribution:
| X = xi : | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P (X = xi) : | 0 | 2 p | 2 p | 3 p | p2 | 2 p2 | 7 p2 | 2 p |
The value of p 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 .
Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.
From a lot of 15 bulbs which include 5 defective, 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.
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
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.
The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class :
(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )
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 random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)
If p : It is a day time , q : It is warm
Give the verbal statements for the following symbolic statements :
(a) p ∧ ∼ q (b) p v q (c) p ↔ q
Find the premium on a property worth ₹12,50,000 at 3% if the property is fully insured.
Verify whether the following function can be regarded as probability mass function (p.m.f.) for the given values of X :
| X | -1 | 0 | 1 |
| P(X = x) | -0.2 | 1 | 0.2 |
The probability that a bomb dropped from an aeroplane will strike a target is `1/5`, If four bombs are dropped, find the probability that :
(a) exactly two will strike the target,
(b) at least one will strike the target.
An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?
The p.d.f. of a continuous r.v. X is given by
f (x) = `1/ (2a)` , for 0 < x < 2a and = 0, otherwise. Show that `P [X < a/ 2] = P [X >( 3a)/ 2]` .
The p.d.f. of r.v. of X is given by
f (x) = `k /sqrtx` , for 0 < x < 4 and = 0, otherwise. Determine k .
Determine c.d.f. of X and hence P (X ≤ 2) and P(X ≤ 1).
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 |
Determine whether each of the following is a probability distribution. Give reasons for your answer.
| y | –1 | 0 | 1 |
| P(y) | 0.6 | 0.1 | 0.2 |
Find the probability distribution of the number of successes in two tosses of a die if success is defined as getting a number greater than 4.
A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. 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. Find the probability distribution of number of tails in two tosses.
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
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?
In a multiple choice test with three possible answers for each of the five questions, what is the probability of a candidate getting four or more correct answers by random choice?
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
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
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 non-negative
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 :
The probability that a bomb will hit the target is 0.8. Find the probability that, out of 5 bombs, exactly 2 will miss the target.
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 0
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.
Find the probability distribution of the number of doublets in three throws of a pair of dice
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.
Two probability distributions of the discrete random variable X and Y are given below.
| X | 0 | 1 | 2 | 3 |
| P(X) | `1/5` | `2/5` | `1/5` | `1/5` |
| Y | 0 | 1 | 2 | 3 |
| P(Y) | `1/5` | `3/10` | `2/10` | `1/10` |
Prove that E(Y2) = 2E(X).
Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.
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
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: The value of A if E(X) = 2.94
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)
The probability distribution of a discrete random variable X is given below:
| X | 2 | 3 | 4 | 5 |
| P(X) | `5/"k"` | `7/"k"` | `9/"k"` | `11/"k"` |
The value of k is ______.
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.
If the p.m.f of a r. v. X is
P(x) = `c/x^3`, for x = 1, 2, 3
= 0, otherwise
then E(X) = ______.
Find the mean number of defective items in a sample of two items drawn one-by-one without replacement from an urn containing 6 items, which include 2 defective items. Assume that the items are identical in shape and size.
Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a nonprime number. The probability that the card was drawn from Box I is ______.
A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.
