Advertisements
Advertisements
प्रश्न
Given the p.d.f. of a continuous r.v. X ,
f (x) = `x^2/3` , for –1 < x < 2 and = 0 otherwise
Determine c.d.f. of X hence find P(1 < x < 2)
Advertisements
उत्तर
Then F(x) = ` int_(-∞)^x f (x) dx`
=` int_(-∞)^-1 f (x) dx + int_(-1)^x f (x) dx`
= 0 + `int_(-1)^x x^2/3 dx = 1/3int_(-1)^x x^2 dx`
= `1/3[x^3/3]_-1^x`
= `1/3[x^3/3-(-1/3)]`
∴ f(x) = `(x^3+1)/9`
P (1 < x < 2) = F (2) - F (1)
= `((2^3+1)/9)- ( (1^3+1)/9)= 1-2/9=7/9`
APPEARS IN
संबंधित प्रश्न
Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f.
f (x) = k `(4 – x^2 )`, for –2 ≤ x ≤ 2 and = 0 otherwise.
P(x > 0)
Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f.
f (x) = k `(4 – x^2)`, for –2 ≤ x ≤ 2 and = 0 otherwise.
P (–0·5 < x or x > 0·5)
The following is the p.d.f. of continuous r.v.
f (x) = `x/8` , for 0 < x < 4 and = 0 otherwise.
Find F(x) at x = 0·5 , 1.7 and 5
Solve the following :
Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.
An economist is interested the number of unemployed graduate in the town of population 1 lakh.
Solve the following problem :
A player tosses two coins. He wins ₹ 10 if 2 heads appear, ₹ 5 if 1 head appears, and ₹ 2 if no head appears. Find the expected value and variance of winning amount.
Fill in the blank :
The value of continuous r.v. are generally obtained by _______
Solve the following problem :
Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
An economist is interested in knowing the number of unemployed graduates in the town with a population of 1 lakh.
Solve the following problem :
Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
Amount of syrup prescribed by a physician.
Solve the following problem :
Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
A person on high protein diet is interested in the weight gained in a week.
Solve the following problem :
Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
Twelve of 20 white rats available for an experiment are male. A scientist randomly selects 5 rats and counts the number of female rats among them.
Solve the following problem :
Identify the random variable as discrete or continuous in each of the following. Identify its range if it is discrete.
A highway safety group is interested in the speed (km/hrs) of a car at a check point.
A coin is tossed 10 times. The probability of getting exactly six heads is ______.
A random variable X has the following probability distribution:
| X = x | 0 | 1 | 2 | 3 |
| P (X = x) | `1/10` | `1/2` | `1/5` | k |
Then the value of k is
A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find P(X ≥ 6)
Find the probability mass function and cumulative distribution function of a number of girl children in families with 4 children, assuming equal probabilities for boys and girls
Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by
`f(x) = {{:((x^2 + 1)/k"," "for" x = 0"," 1"," 2),(0"," "otherwise"):}`
Find the value of k
Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by
`f(x) = {{:((x^2 + 1)/k"," "for" x = 0"," 1"," 2),(0"," "otherwise"):}`
Find cumulative distribution function
Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by
`f(x) = {{:((x^2 + 1)/k"," "for" x = 0"," 1"," 2),(0"," "otherwise"):}`
Find P(X ≥ 1)
The cumulative distribution function of a discrete random variable is given by
F(x) = `{{:(0, "for" - oo < x < 0),(1/2, "for" 0 ≤ x < 1),(3/5, "for" 1 ≤ x < 2),(4/5, "for" 2 ≤ x < 4),(9/5, "for" 3 ≤ x < 4),(1, "for" ≤ x < oo):}`
Find P(X < 3)
The cumulative distribution function of a discrete random variable is given by
F(x) = `{{:(0, "for" - oo < x < 0),(1/2, "for" 0 ≤ x < 1),(3/5, "for" 1 ≤ x < 2),(4/5, "for" 2 ≤ x < 4),(9/5, "for" 3 ≤ x < 4),(1, "for" ≤ x < oo):}`
Find P(X ≥ 2)
If Xis a.r.v. with c.d.f F (x) and its probability distribution is given by
| X = x | - 1.5 | -0.5 | 0.5 | 1.5 | 2.5 |
| P(X = x) | 0.05 | 0.2 | 0.15 | 0.25 | 0.35 |
then, F(1.5) - F(- 0.5) = ?
Choose the correct alternative:
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
Choose the correct alternative:
Which of the following is a discrete random variable?
I. The number of cars crossing a particular signal in a day.
II. The number of customers in a queue to buy train tickets at a moment.
III. The time taken to complete a telephone call.
If A = {x ∈ R : x2 - 5 |x| + 6 = 0}, then n(A) = _____.
If the probability function of a random variable X is defined by P(X = k) = a`((k + 1)/2^k)` for k - 0, 1, 2, 3, 4, 5, then the probability that X takes a prime value is ______
A random variable X has the following probability distribution:
| X | 1 | 2 | 3 | 4 |
| P(X) | `1/3` | `2/9` | `1/3` | `1/9` |
1hen, the mean of this distribution is ______
The probability distribution of a random variable X is given below. If its mean is 4.2, then the values of a and bar respectively
| X = x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | a | a | a | b | b | 0.3 |
The probability distribution of a random variable X is given below.
| X = k | 0 | 1 | 2 | 3 | 4 |
| P(X = k) | 0.1 | 0.4 | 0.3 | 0.2 | 0 |
The variance of X is ______
A card is chosen from a well-shuffled pack of cards. The probability of getting an ace of spade or a jack of diamond is ______.
Two coins are tossed. Then the probability distribution of number of tails is.
The c.d.f. of a discrete r.v. X is
| X = x | -4 | -2 | -1 | 0 | 2 | 4 | 6 | 8 |
| F(x) | 0.2 | 0.4 | 0.55 | 0.6 | 0.75 | 0.80 | 0.95 | 1 |
Then P(X ≤ 4|X > -1) = ?
The p.d.f. of a continuous random variable X is
f(x) = 0.1 x, 0 < x < 5
= 0, otherwise
Then the value of P(X > 3) is ______
A coin is tossed three times. If X denotes the absolute difference between the number of heads and the number of tails then P(X = 1) = ______.
At random variable X – B(n, p), if values of mean and variance of X are 18 and 12 respectively, then total number of possible values of X are ______.
For the following distribution function F(x) of a rv.x.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| F(x) | 0.2 | 0.37 | 0.48 | 0.62 | 0.85 | 1 |
P(3 < x < 5) =
If f(x) = `k/2^x` is a probability distribution of a random variable X that can take on the values x = 0, 1, 2, 3, 4. Then, k is equal to ______.
