Advertisements
Advertisements
प्रश्न
Find the mean number of heads in three tosses of a fair coin.
उत्तर
Let X denote the success of getting heads.
Therefore, the sample space is
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
It can be seen that X can take the value of 0, 1, 2 or 3
∴ P(X = 0) = P(TTT)
= P(T) . P(T) . P(T)
= `1/2 xx 1/2 xx 1/2`
= `1/8`
∴ P(X = 1) = P(HHT) + P(HTH) + P(THH)
= `1/2 xx 1/2 xx 1/2 + 1/2 xx 1/2 xx 1/2 + 1/2 xx 1/2 xx 1/2`
= `3/8`
∴ P(X = 2) = P(HHT) + P(HTH) + P(THH)
`1/2 xx 1/2 xx 1/2 + 1/2 xx 1/2 xx 1/2 + 1/2 xx 1/2 xx 1/2`
= `3/8`
∴ P(X = 3) = P(HHH)
= `1/2 xx 1/2 xx 1/2`
= `1/8`
Therefore, the required probability distribution is as follows.
| X | 0 | 1 | 2 | 3 |
| P(X) | `1/8` | `3/8` | `3/8` | `1/8` |
Mean of X E(X), µ =`sum X_iP(X_i)`
= `0 xx1/8 + 1xx3/8 + 2xx3/8 + 3xx1/8`
= `0 + 3/8 + 3/4 + 3/8`
= `12/8`
= 1.5
APPEARS IN
संबंधित प्रश्न
State if the following is not the probability mass function of a random variable. Give reasons for your answer.
| X | 0 | 1 | 2 |
| P(X) | 0.4 | 0.4 | 0.2 |
State if the following is not the probability mass function 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 | 0.2 |
State if the following is not the probability mass function of a random variable. Give reasons for your answer
| Z | 3 | 2 | 1 | 0 | −1 |
| P(Z) | 0.3 | 0.2 | 0.4 | 0 | 0.05 |
State if the following is not the probability mass function of a random variable. Give reasons for your answer.
| X | 0 | -1 | -2 |
| P(X) | 0.3 | 0.4 | 0.3 |
The following is the p.d.f. of r.v. X :
f(x) = `x/8`, for 0 < x < 4 and = 0 otherwise
P ( 1 < x < 2 )
The following is the p.d.f. of r.v. X:
f(x) = `x/8`, for 0 < x < 4 and = 0 otherwise.
P(x > 2)
If a r.v. X has p.d.f.,
f (x) = `c /x` , for 1 < x < 3, c > 0, Find c, E(X) and Var (X).
Solve the following :
Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.
Amount of syrup prescribed by physician.
Solve the following :
The following probability distribution of r.v. X
| 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 :
A fair coin is tossed 4 times. Let X denote the number of heads obtained. Identify the probability distribution of X and state the formula for p. m. f. of X.
The probability distribution of discrete r.v. X is as follows :
| x = x | 1 | 2 | 3 | 4 | 5 | 6 |
| P[x=x] | k | 2k | 3k | 4k | 5k | 6k |
(i) Determine the value of k.
(ii) Find P(X≤4), P(2<X< 4), P(X≥3).
Let X be amount of time for which a book is taken out of library by randomly selected student and suppose X has p.d.f
f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.
Calculate: P(x≤1)
Find the probability distribution of number of number of tails in three tosses of a coin
Given that X ~ B(n,p), if n = 25, E(X) = 10, find p and Var (X).
F(x) is c.d.f. of discrete r.v. X whose distribution is
| Xi | – 2 | – 1 | 0 | 1 | 2 |
| Pi | 0.2 | 0.3 | 0.15 | 0.25 | 0.1 |
Then F(– 3) = _______ .
Choose the correct alternative :
X: is number obtained on upper most face when a fair die….thrown then E(X) = _______.
If F(x) is distribution function of discrete r.v.x with p.m.f. P(x) = `(x - 1)/(3)` for x = 0, 1 2, 3, and P(x) = 0 otherwise then F(4) = _______.
Fill in the blank :
E(x) is considered to be _______ of the probability distribution of x.
State whether the following is True or False :
| x | – 2 | – 1 | 0 | 1 | 2 |
| P(X = x) | 0.2 | 0.3 | 0.15 | 0.25 | 0.1 |
If F(x) is c.d.f. of discrete r.v. X then F(–3) = 0
If r.v. X assumes values 1, 2, 3, ……. n with equal probabilities then E(X) = `("n" + 1)/(2)`
Solve the following problem :
Find the expected value and variance of the r. v. X if its probability distribution is as follows.
| x | 1 | 2 | 3 | ... | n |
| P(X = x) | `(1)/"n"` | `(1)/"n"` | `(1)/"n"` | ... | `(1)/"n"` |
Solve the following problem :
Let X∼B(n,p) If n = 10 and E(X)= 5, find p and Var(X).
If a d.r.v. X has the following probability distribution:
| X | –2 | –1 | 0 | 1 | 2 | 3 |
| P(X = x) | 0.1 | k | 0.2 | 2k | 0.3 | k |
then P(X = –1) is ______
Choose the correct alternative:
f(x) is c.d.f. of discete r.v. X whose distribution is
| xi | – 2 | – 1 | 0 | 1 | 2 |
| pi | 0.2 | 0.3 | 0.15 | 0.25 | 0.1 |
then F(– 3) = ______
If p.m.f. of r.v. X is given below.
| x | 0 | 1 | 2 |
| P(x) | q2 | 2pq | p2 |
then Var(x) = ______
If X is discrete random variable takes the values x1, x2, x3, … xn, then `sum_("i" = 1)^"n" "P"(x_"i")` = ______
E(x) is considered to be ______ of the probability distribution of x.
The probability distribution of a discrete r.v.X is as follows.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | k | 2k | 3k | 4k | 5k | 6k |
Complete the following activity.
Solution: Since `sum"p"_"i"` = 1
k = `square`
The probability distribution of a discrete r.v.X is as follows.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | k | 2k | 3k | 4k | 5k | 6k |
Complete the following activity.
Solution: Since `sum"p"_"i"` = 1
P(X ≤ 4) = `square + square + square + square = square`
The probability distribution of a discrete r.v. X is as follows:
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = x) | k | 2k | 3k | 4k | 5k | 6k |
- Determine the value of k.
- Find P(X ≤ 4)
- P(2 < X < 4)
- P(X ≥ 3)
If F(x) is distribution function of discrete r.v.x with p.m.f. P(x) = `(x - 1)/(3)`; for x = 0, 1 2, 3, and P(x) = 0 otherwise then F(4) = _______.
The value of discrete r.v. is generally obtained by counting.
