Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let P (X = 0) = k. Then,
P (X = 0) = P (X > 0) = P (X < 0)
P (X < 0) = k ∴ P (X = 0) + P (X > 0) + P (X < 0) = 1
\[\Rightarrow k + k + k = 1\]
\[ \Rightarrow k = \frac{1}{3}\]
Now,
P (X < 0) = k
\[ \Rightarrow 3P\left( X = - 1 \right) = k ................ \left [ \because P\left( X = - 1 \right) = P\left( X = - 2 \right) = P\left( X = - 3 \right) \right]\]
\[ \Rightarrow P\left( X = - 1 \right) = \frac{k}{3}\]
\[ \Rightarrow P\left( X = - 1 \right) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\]
\[ \therefore P\left( X = - 1 \right) = P\left( X = - 2 \right) = P\left( X = - 3 \right) = \frac{1}{9}\]
\[\text{ Similarly,} \]
\[P\left( X > 0 \right) = k\]
\[ \Rightarrow P\left( X = 1 \right) = P\left( X = 2 \right) = P\left( X = 3 \right) = \frac{1}{9}\]
| Xi | Pi |
| -3 |
\[\frac{1}{9}\]
|
| -2 |
\[\frac{1}{9}\]
|
| -1 |
\[\frac{1}{9}\]
|
| 1 |
\[\frac{1}{9}\]
|
| 2 |
\[\frac{1}{9}\]
|
| 3 |
\[\frac{1}{9}\]
|
APPEARS IN
संबंधित प्रश्न
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
An urn contains 25 balls of which 10 balls bear a mark ‘X’ and the remaining 15 bear a mark ‘Y’. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that
(i) all will bear ‘X’ mark.
(ii) not more than 2 will bear ‘Y’ mark.
(iii) at least one ball will bear ‘Y’ mark
(iv) the number of balls with ‘X’ mark and ‘Y’ mark will be equal.
A random variable X has the following probability distribution:
| Values of X : | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P (X) : | a | 3a | 5a | 7a | 9a | 11a | 13a | 15a | 17a |
Determine:
(i) The value of a
(ii) P (X < 3), P (X ≥ 3), P (0 < X < 5).
Let X be a random variable which assumes values x1, x2, x3, x4 such that 2P (X = x1) = 3P(X = x2) = P (X = x3) = 5 P (X = x4). Find 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.
Five defective mangoes are accidently mixed with 15 good ones. Four mangoes are drawn at random from this lot. Find the probability distribution of the number of defective mangoes.
Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.
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.
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 | 2 | 3 | 4 |
| pi : | 0.4 | 0.3 | 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 coin is tossed four times. Let X denote the number of heads occurring. 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 random variable has the following probability distribution:
| X = xi : | 1 | 2 | 3 | 4 |
| P (X = xi) : | k | 2k | 3k | 4k |
Write the value of P (X ≥ 3).
Mark the correct alternative in the following question:
Let X be a discrete random variable. Then the variance of X is
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.
Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).
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.
Write the negation of the following statements :
(a) Chetan has black hair and blue eyes.
(b) ∃ x ∈ R such that x2 + 3 > 0.
If X ∼ N (4,25), then find P(x ≤ 4)
Find the premium on a property worth ₹12,50,000 at 3% if the property is fully insured.
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.
A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two 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?
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 :
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.
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 |
Calculate `"V"("X"/2)`
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.
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 P(X ≤ 2) and P(X > 2)
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)
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.
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.
Two numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of two numbers selected, find the probability distribution of X.
Kiran plays a game of throwing a fair die 3 times but to quit as and when she gets a six. Kiran gets +1 point for a six and –1 for any other number.
- If X denotes the random variable “points earned” then what are the possible values X can take?
- Find the probability distribution of this random variable X.
- Find the expected value of the points she gets.
