Advertisements
Advertisements
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
Advertisements
उत्तर
By definition cummulative distribution function at x is
`P(x ≥ 2) = 0.3 + 0.4 + 0.2 = 0.9`
`f (x_i) = P_1 + P_2 + P_3 + ……. + P_i` where, i = 1, …, x
Thus `f(x_1)=P_1=0.1`
`f(x_2)=P_1=0.1`
`f(x_2)=P_1+P_2=0.1+0.3=0.4`
`f(x_3)=P_1+P_2+P_3=0.1+0.3+0.4=0.8`
`f(x_4)=P_1+P_2+P_3+P_4=0.1+0.3+0.4+0.2=1`
`therefore f(x_4)=sum_(i=1)^4P_i=1`
| X = x | 1 | 2 | 3 | 4 |
| P(X = x) | 0.1 | 0.4 | 0.8 | 1 |
APPEARS IN
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.
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 |
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.
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.
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).
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 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?
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
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.
If X denotes the number on the upper face of a cubical die when it is thrown, find the mean of X.
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
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.
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?
From the following data, find the crude death rates (C.D.R.) for Town I and Town II, and comment on the results :
| Age Group (in years) | Town I | Town II | ||
| Population | No. of deaths | Population | No. of deaths | |
| 0-10 | 1500 | 45 | 6000 | 150 |
| 10-25 | 5000 | 30 | 6000 | 40 |
| 25 - 45 | 3000 | 15 | 5000 | 20 |
| 45 & above | 500 | 22 | 3000 | 54 |
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 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).
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.
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 = 0
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
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
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 :
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.
Find the mean and variance of the number randomly selected from 1 to 15
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as six appears on at least one die
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 biased dice are thrown together. For the first die P(6) = `1/2`, the other scores being equally likely while for the second die, P(1) = `2/5` and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’.
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)
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 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 ______.
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.
