Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Let X be the random variable denotes the number of girl child among 4 children
X = {0, 1, 2, 3, 4}
| Values of the random variable | 0 | 1 | 2 | 3 | 4 | Total |
| Number of elements in inverse image | 1 | 4 | 6 | 4 | 1 | 16 |
(i) Probability mass function
| x | 0 | 1 | 2 | 3 | 4 | Total |
| f(x) | `1/16` | `4/16` | `6/16` | `4/16` | `1/16` | 1 |
(ii) Cumulative distribution
F(x) = P(X ≤ x)
= `sum_(x_"i" ≤ x) "P"("X" = x_"i")`
P(X < 0) = 0 for `- oo < x < 0`
F(0) = P(X ≤ 0)
= `P(X = 0)
= `1/16`
F(1) = P(X ≤ 1) = P(X = 0) + P(X = 1)`
= `1/16 + 4/16`
= `5/16`
F(2) = P(X ≤ 2)
= P(X = 0) + P(X = 1) + P(X = 2)
= `1/16 + 4/16 + 6/16`
= `11/16`
F(3) = P(X ≤ 3)
= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
= `1/16 + 4/16 + 6/16 + 4/16`
= `15/16`
F(4) = P(X ≤ 4)
= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
= `15/16 + 1/16`
= 1
F(x) = `{{:(0",", "For" x < 0),(1/16",", "For" x ≤ 0),(5/16",", "For" x ≤ 1), (11/16",", "For" x ≤ 2),(15/16",", "For" x ≤ 3),(1",", "For" x ≤ 4):}`
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 (–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
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)
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 :
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.
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
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, - oo < x < - 1),(0.15, - 1 ≤ x < 0),(0.35, 0 ≤ x < 1),(0.60, 1 ≤ x < 2),(0.85, 2 ≤ x < 3),(1, 3 ≤ x < oo):}`
Find P(X < 1)
A random variable X has the following probability mass function.
| x | 1 | 2 | 3 | 4 | 5 |
| F(x) | k2 | 2k2 | 3k2 | 2k | 3k |
Find P(2 ≤ X < 5)
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)
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:
Suppose that X takes on one of the values 0, 1 and 2. If for some constant k, P(X = i) = kP(X = i – 1) for i = 1, 2 and P(X = 0) = `1/7`. Then the value of k is
Let X = time (in minutes) that lapses between the ringing of the bell at the end of a lecture and the actual time when the professor ends the lecture. Suppose X has p.d.f.
f(x) = `{(kx^2"," 0 ≤ x ≤ 2), (0"," "othenwise"):}`
Then, the probability that the lecture ends within 1 minute of the bell ringing is ______
If A = {x ∈ R : x2 - 5 |x| + 6 = 0}, then n(A) = _____.
X is a continuous random variable with a probability density function
f(x) = `{{:(x^2/4 + k; 0 ≤ x ≤ 2),(0; "otherwise"):}`
The value of k is equal to ______
Two cards are randomly drawn, with replacement. from a well shuffled deck of 52 playing cards. Find the probability distribution of the number of aces drawn.
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 ______.
