Advertisements
Advertisements
Question
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).
Solution
Since, f is p.d.f. of the r.v. X,
` int_(-∞)^∞ f(x) dx = 1`
∴ `int_(-∞)^∞ f(x) dx + int_(-0)^4 f(x) dx + int_(4)^∞ f(x) dx = 1`
∴`0 + int_(0)^4 k /sqrtx dx + 0 = 1`
∴ `k int_(0)^4 x ^(-1/2) dx = 1`
∴ `k[x^(1/2)/(1/2)]_0^4 =1`
∴ 2k[2 - 0] = 1
∴ 4k = 1
∴ k = `1/4`
Let F(X) be the c.d.f. of X.
∴ F(X) = P(X ≤ x) = `int_(-∞)^x f(x) dx`
= `int_(-∞)^0 f (x) dx + int_(0)^x f(x) dx`
= `0 + int_(0)^x k /sqrtx dx`
= `k int_(0)^x x ^(-1/2) dx`
= `k[x^(1/2)/(1/2)]_0^x`
= `2k sqrtx = 2(1/4) sqrtx` .....[∵ k = `1/4`]
∴ F(X) = `sqrtx /2`
P(X ≤ 2) = F (2) = `sqrt2/2 = 1/sqrt2`
P(X ≤ 1) = F (1) = `sqrt1/2 = 1/2`
Hence, `k 1/ 4 ,P (X ≤ 2) = 1/sqrt2 , P (X ≤ 1) = 1/2`
APPEARS IN
RELATED QUESTIONS
From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain -
(a) exactly 1 defective bulb.
(b) at least 1 defective bulb.
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 four tosses of a coin.
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die
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 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?
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.
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)
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.
Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.
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.
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 Y in two throws of two dice, where Y represents the number of times a total of 9 appears.
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.
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 P(X ≤ 2) and P(X > 2) .
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) .
Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.
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 |
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 : | -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 |
Find the value of k.
Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.
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 .
For what value of k the following distribution is a probability distribution?
| X = xi : | 0 | 1 | 2 | 3 |
| P (X = xi) : | 2k4 | 3k2 − 5k3 | 2k − 3k2 | 3k − 1 |
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).
Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.
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.
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.
For the following probability density function (p. d. f) of X, find P(X < 1) and P(|x| < 1)
`f(x) = x^2/18, -3 < x < 3`
= 0, otherwise
Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).
Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).
The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class :
(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )
Find mean and standard deviation of the continuous random variable X whose p.d.f. is given by f(x) = 6x(1 - x);= (0); 0 < x < 1(otherwise)
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?
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.
A fair coin is tossed 12 times. Find the probability of getting exactly 7 heads .
A fair coin is tossed 12 times. Find the probability of getting at least 2 heads .
Write the negation of the following statements :
(a) Chetan has black hair and blue eyes.
(b) ∃ x ∈ R such that x2 + 3 > 0.
The expenditure Ec of a person with income I is given by Ec = (0.000035) I2 + (0. 045) I. Find marginal propensity to consume (MPC) and average propensity to consume (APC) when I = 5000.
Find the premium on a property worth ₹12,50,000 at 3% if the property is fully insured.
The p.m.f. of a random variable X is
`"P"(x) = 1/5` , for x = I, 2, 3, 4, 5
= 0 , otherwise.
Find E(X).
A card is drawn at random and replaced four times from a well shuftled pack of 52 cards. Find the probability that -
(a) Two diamond cards are drawn.
(b) At least one diamond card is drawn.
Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.
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).
A coin is biased so that the head is 3 times as likely to occur as tail. Find the probability distribution of number of tails in two tosses.
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
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
In a multiple choice test with three possible answers for each of the five questions, what is the probability of a candidate getting four or more correct answers by random choice?
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.
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 1 terminal requires attention during a week.
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 ______
Find the probability distribution of the number of doublets in three throws of a pair of dice
A discrete random variable X has the probability distribution given as below:
| X | 0.5 | 1 | 1.5 | 2 |
| P(X) | k | k2 | 2k2 | k |
Determine the mean of the distribution.
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 below:
| X | 0 | 1 | 2 | 3 |
| P(X) | k | `"k"/2` | `"k"/4` | `"k"/8` |
Find P(X ≤ 2) + P (X > 2)
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
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 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 E(X)
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.
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 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 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.
