Advertisements
Advertisements
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.
उत्तर
Total number of bulbs = 15
Number of defective bulbs, i.e., n(D) = 5
`∴ P(D) = 5/15=1/3`
Number of non-defective bulbs, n(ND) = 10
`∴ P(ND) = 10/15=2/3`
Let X be a random variable that shows the number of defective bulbs in a draw of 4 bulbs.
Clearly, X can take the values 0, 1, 2, 3 and 4.
P(X = 0) = P (no defective bulbs)
` = (2/3)^4=16/81`
P(X = 1) = P (1 defective and 3 non-defective bulbs)
`= 4 xx1/3xx(2/3)^3=32/81`
P(X = 2) = P (2 defective and 2 non-defective bulbs)
`= 6xx(1/3)^2xx(2/3)^2=24/81`
P(X = 3) = P (3 defective and 1 non-defective bulb)
` = 4xx (1/3)^3xx 2/3=8/81`
P(X = 4) = P (All bulbs are defective)
` = (1/3)^4=1/81`
Now, probability distribution is given by
| X | 0 | 1 | 2 | 3 | 4 |
| pi |
`16/81` |
`32/81` |
`24/81` |
`8/81` |
`1/81` |
`∴ Mean = ∑_i x_i p_i = 0xx16/81 + 1xx32/81 + 2xx24/81 + 3xx8/81 + 4xx1/81`
`= 0 + 32/81 + 48/81 + 24/81 + 4/81 `
`=108/81=4/3`
Hence, the mean of the distribution is `4/3.`
APPEARS IN
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
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?
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 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
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 |
Determine the mean of the distribution.
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 1 or 3 according as an odd or an even number appears. Find the probability distribution, mean and variance of X.
Find the mean of the following probability distribution:
| X= xi: | 1 | 2 | 3 |
| P(X= xi) : |
\[\frac{1}{4}\]
|
\[\frac{1}{8}\]
|
\[\frac{5}{8}\]
|
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 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
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
A fair coin is tossed 12 times. Find the probability of getting exactly 7 heads .
The following table gives the age of the husbands and of the wives :
| Age of wives (in years) |
Age of husbands (in years) |
|||
| 20-30 | 30- 40 | 40- 50 | 50- 60 | |
| 15-25 | 5 | 9 | 3 | - |
| 25-35 | - | 10 | 25 | 2 |
| 35-45 | - | 1 | 12 | 2 |
| 45-55 | - | - | 4 | 16 |
| 55-65 | - | - | - | 4 |
Find the marginal frequency distribution of the age of husbands.
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]
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 |
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.
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
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 :
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.
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
Let X be a discrete random variable. The probability distribution of X is given below:
| X | 30 | 10 | – 10 |
| P(X) | `1/5` | `3/10` | `1/2` |
Then E(X) is equal to ______.
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 probability distribution of a random variable X is given below:
| X | 0 | 1 | 2 | 3 |
| P(X) | k | `"k"/2` | `"k"/4` | `"k"/8` |
Find P(X ≤ 2) + P (X > 2)
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 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(X)
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 ______.
