Advertisements
Advertisements
प्रश्न
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)
उत्तर
Given p.d.f is
f(x) = 6x(l - x), 0 < x< 1
= 0 otherwise
Mean = E(X)
= `∫_0^1 x f(x)` dx
= `∫_0^1 x (6x - 6x^2)` dx
= `∫_0^1 6x^2 dx - 6 ∫_0^1 x^3 dx`
= `6[x^3/3]_0^1 - 6[x^4/4]_0^1`
= `6[1/3] - 6[1/4]`
∴ Mean = `2 - 3/2 = 0.5` ....(i)
`"Var" ("X") = "E"("X"^2) - ["E"("X")]^2`
= `∫_0^1 x^2 f(x) dx - (0.5)^2` .....[by (i)]
= `∫_0^1 x^2(6x - 6x^2) dx - (0.5)^2`
= `6∫_0^1 x^3 dx - 6 ∫_0^1 x^4 dx - 0.25`
= `6[x^4/4]_0^1 - 6[x^5/5]_0^1 - 0.25`
= `6[1/4] - 6[1/5] - 0.25`
= 0.3 - 0.25
Var.(X) = 0.05
S.D. (X) = `sqrt("Var".("X"))`
= `sqrt(0.05)`
S.D.(X) = 0.2236
APPEARS IN
संबंधित प्रश्न
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 |
| 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.
| Z | 3 | 2 | 1 | 0 | -1 |
| P(Z) | 0.3 | 0.2 | 0.4 | 0.1 | 0.05 |
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?
Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X
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)
Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of number of successes.
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 .
Find the mean and standard deviation of each of the following probability distribution:
| xi : | −1 | 0 | 1 | 2 | 3 |
| pi : | 0.3 | 0.1 | 0.1 | 0.3 | 0.2 |
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.
A box contains 13 bulbs, out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement, from the box. Find the probability distribution of the number of defective bulbs.
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).
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
Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.
Calculate `"e"_0^circ ,"e"_1^circ , "e"_2^circ` from the following:
| Age x | 0 | 1 | 2 |
| lx | 1000 | 880 | 876 |
| Tx | - | - | 3323 |
If the demand function is D = 150 - p2 - 3p, find marginal revenue, average revenue and elasticity of demand for price p = 3.
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 |
If p : It is a day time , q : It is warm
Give the verbal statements for the following symbolic statements :
(a) p ∧ ∼ q (b) p v q (c) p ↔ q
The probability that a bomb dropped from an aeroplane will strike a target is `1/5`, If four bombs are dropped, find the probability that :
(a) exactly two will strike the target,
(b) at least one will strike the target.
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?
Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.
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 |
Find the probability distribution of the number of successes in two tosses of a die if success is defined as getting a number greater than 4.
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.
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes
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
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 :
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 positive.
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.
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
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 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 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.
