Advertisements
Advertisements
प्रश्न
Determine whether each of the following is a probability distribution. Give reasons for your answer.
| x | 0 | 1 | 2 |
| P(x) | 0.1 | 0.6 | 0.3 |
उत्तर
Here, P(x) > 0 for all values of x
Now consider,
`sumP(x)` = 0.1 + 0.6 + 0.3 = 1
∴ Given distribution is a probability distribution.
संबंधित प्रश्न
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.
| X | 0 | 1 | 2 | 3 | 4 |
| P(X) | 0.1 | 0.5 | 0.2 | -0.1 | 0.3 |
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represents the number of black balls. What are the possible values of X? Is X a random variable?
Find the probability distribution of number of heads in two tosses of a coin.
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denotes the larger of the two numbers obtained. Find E(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?
If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)19 = 0⋅1348]
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)
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)
Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
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.
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 : | −1 | 0 | 1 | 2 | 3 |
| pi : | 0.3 | 0.1 | 0.1 | 0.3 | 0.2 |
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 |
Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
A fair die is tossed. Let X denote twice the number appearing. Find the probability distribution, mean and variance of X.
A fair coin is tossed four times. Let X denote the longest string of heads occurring. Find the probability distribution, mean and variance of X.
A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.
Three cards are drawn at random (without replacement) from a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence, find the mean of the distribution .
An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X ? Is X a random variable ? If yes, then find the 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 is a random-variable with probability distribution as given below:
| X = xi : | 0 | 1 | 2 | 3 |
| P (X = xi) : | k | 3 k | 3 k | k |
The value of k and its variance are
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 .
John and Mathew started a business with their capitals in the ratio 8 : 5. After 8 months, john added 25% of his earlier capital as further investment. At the same time, Mathew withdrew 20% of bis earlier capital. At the end of the year, they earned ₹ 52000 as profit. How should they divide the profit between them?
Write the negation of the following statements :
(a) Chetan has black hair and blue eyes.
(b) ∃ x ∈ R such that x2 + 3 > 0.
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.
Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.
Solve the following :
Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.
20 white rats are available for an experiment. Twelve rats are male. Scientist randomly selects 5 rats number of female rats selected on a specific day
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.
| x | 0 | 1 | 2 | 3 | 4 |
| P(x) | 0.1 | 0.5 | 0.2 | –0.1 | 0.3 |
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.
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?
The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of X > 1
10 balls are marked with digits 0 to 9. If four balls are selected with replacement. What is the probability that none is marked 0?
Solve the following problem :
The probability that a lamp in the classroom will burn is 0.3. 3 lamps are fitted in the classroom. The classroom is unusable if the number of lamps burning in it is less than 2. Find the probability that the classroom cannot be used on a random occasion.
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.
Find the probability that the visitor obtains the answer yes from at least 3 students.
Let the p.m.f. of a random variable X be P(x) = `(3 - x)/10`, for x = −1, 0, 1, 2 = 0, otherwise Then E(x) is ______
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.
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 |
Variance of X.
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 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 E(3X2)
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 ______.
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.
Find the probability distribution of the number of successes in two toves of a die where a success is define as:- Six appeared on at least one die.
A random variable X has the following probability distribution:
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P(x) | k | 2k | 2k | 3k | k2 | 2k2 | 7k2 + k |
Find:
- k
- P(X < 3)
- P(X > 4)
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`
