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प्रश्न
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 |
उत्तर
Here, P(X = 3) = –0.1 < 0
Probability for an value of x cannot be negative.
∴ Given distribution is not a probability distribution.
संबंधित प्रश्न
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 |
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.
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.
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 |
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)
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?
Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.
From a lot containing 25 items, 5 of which are defective, 4 are chosen at random. Let X be the number of defectives found. Obtain the probability distribution of X if the items are chosen without replacement .
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.
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.
The probability distribution of a random variable X is given below:
| x | 0 | 1 | 2 | 3 |
| P(X) | k |
\[\frac{k}{2}\]
|
\[\frac{k}{4}\]
|
\[\frac{k}{8}\]
|
Determine the value of k .
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) .
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 : | -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 : | -3 | -1 | 0 | 1 | 3 |
| pi : | 0.05 | 0.45 | 0.20 | 0.25 | 0.05 |
Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.
Write the values of 'a' for which the following distribution of probabilities becomes a probability distribution:
| X= xi: | -2 | -1 | 0 | 1 |
| P(X= xi) : |
\[\frac{1 - a}{4}\]
|
\[\frac{1 + 2a}{4}\]
|
\[\frac{1 - 2a}{4}\]
|
\[\frac{1 + a}{4}\]
|
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
Mark the correct alternative in the following question:
Let X be a discrete random variable. Then the variance of X is
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes and, hence, find its mean.
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.
Three fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X.
Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).
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 |
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?
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?
A random variable X has the following probability distribution :
| X = x | -2 | -1 | 0 | 1 | 2 | 3 |
| P(x) | 0.1 | k | 0.2 | 2k | 0.3 | k |
Find the value of k and calculate mean.
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.
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]` .
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 |
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 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?
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
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 :
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.
Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X
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.
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).
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
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)
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 to following probability distribution.
| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P(x) | 0 | k | 2k | 2k | 3k | k2 | 2k2 | 7k2 + k |
Determine
A person throws two fair dice. He wins ₹ 15 for throwing a doublet (same numbers on the two dice), wins ₹ 12 when the throw results in the sum of 9, and loses ₹ 6 for any other outcome on the throw. Then the expected gain/loss (in ₹) of the person is ______.
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 primary school teacher wants to teach the concept of 'larger number' to the students of Class II.
To teach this concept, he conducts an activity in his class. He asks the children to select two numbers from a set of numbers given as 2, 3, 4, 5 one after the other without replacement.
All the outcomes of this activity are tabulated in the form of ordered pairs given below:
| 2 | 3 | 4 | 5 | |
| 2 | (2, 2) | (2, 3) | (2, 4) | |
| 3 | (3, 2) | (3, 3) | (3, 5) | |
| 4 | (4, 2) | (4, 4) | (4, 5) | |
| 5 | (5, 3) | (5, 4) | (5, 5) |
- Complete the table given above.
- Find the total number of ordered pairs having one larger number.
- Let the random variable X denote the larger of two numbers in the ordered pair.
Now, complete the probability distribution table for X given below.
X 3 4 5 P(X = x) - Find the value of P(X < 5)
- Calculate the expected value of the probability distribution.
