###### Advertisements

###### Advertisements

**Solve the following problem :**

The probability distribution of a discrete r.v. X is as follows.

X |
1 | 2 | 3 | 4 | 5 | 6 |

(X = x) |
k | 2k | 3k | 4k | 5k | 6k |

Find P(X ≤ 4), P(2 < X < 4), P(X ≤ 3).

###### Advertisements

#### Solution

a. P(X ≤ 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

= k + 2k + 3k + 4k

= 10k

= `(10)/(21)`

b. P(2 < X < 4) = P(X = 3) = 3k = `(3)/(21) = (1)/(7)`

c. P(X ≥ 3) = 1 – P(X < 3)

= 1 – [P(X = 1) + P(X = 2)]

= 1 – (k + 2k) = 1 – 3k

= `1 - (3)/(21)`

= `1 - (1)/(7)`

=`(6)/(7)`.

#### APPEARS IN

#### RELATED QUESTIONS

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.

X |
0 | 1 | 2 |

P(X) |
0.1 | 0.6 | 0.3 |

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 |

Y |
−1 | 0 | 1 |

P(Y) |
0.6 | 0.1 | 0.2 |

**A random variable X has the following probability distribution:**

X |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

P(X) |
0 | k | 2k | 2k | 3k | k^{2} |
2k^{2} |
7k^{2} + k |

**Determine:**

- k
- P(X < 3)
- P( X > 4)

Find the mean number of heads in three tosses of a fair coin.

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of X.

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)

It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by

f (x) = `x^2 /3` , for –1 < x < 2 and = 0 otherwise

Verify whether f (x) is p.d.f. of r.v. X.

Find k if the following function represent p.d.f. of r.v. X

f (x) = kx, for 0 < x < 2 and = 0 otherwise, Also find P `(1/ 4 < x < 3 /2)`.

Find k, if the following function represents p.d.f. of r.v. X.

f(x) = kx(1 – x), for 0 < x < 1 and = 0, otherwise.

Also, find `P(1/4 < x < 1/2) and P(x < 1/2)`.

Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by

f (x) = `1/5` , for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that waiting time is more than 4 minutes.

Choose the correct option from the given alternative:

If the p.d.f of a.c.r.v. X is f (x) = 3 (1 − 2x2 ), for 0 < x < 1 and = 0, otherwise (elsewhere) then the c.d.f of X is F(x) =

**Choose the correct option from the given alternative:**

If p.m.f. of a d.r.v. X is P (X = x) = `((c_(x)^5 ))/2^5` , for x = 0, 1, 2, 3, 4, 5 and = 0, otherwise If a = P (X ≤ 2) and b = P (X ≥ 3), then E (X ) =

**Choose the correct option from the given alternative :**

If p.m.f. of a d.r.v. X is P (x) = `c/ x^3` , for x = 1, 2, 3 and = 0, otherwise (elsewhere) then E (X ) =

**Choose the correct option from the given alternative:**

If the 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) =

**Choose the correct option from the given alternative:**

If the 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) =

**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 following is the c.d.f. of r.v. X

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |

F(X) | 0.1 | 0.3 | 0.5 | 0.65 | 0.75 | 0.85 | 0.9 |
1 |

P (X ≤ 3/ X > 0)

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(0.5 ≤ x ≤ 1.5)

Find the probability distribution of number of number of tails in three tosses of a coin

Find expected value and variance of X, the number on the uppermost face of a fair die.

70% of the members favour and 30% oppose a proposal in a meeting. The random variable X takes the value 0 if a member opposes the proposal and the value 1 if a member is in favour. Find E(X) and Var(X).

Given that X ~ B(n, p), if n = 10 and p = 0.4, find E(X) and Var(X)

Given that X ~ B(n,p), if n = 25, E(X) = 10, find p and Var (X).

The expected value of the sum of two numbers obtained when two fair dice are rolled is _______.

**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 X denotes the number on the uppermost face of cubic die when it is tossed, then E(X) is ______

The p.m.f. of a d.r.v. X is P(X = x) = `{{:(((5),(x))/2^5",", "for" x = 0"," 1"," 2"," 3"," 4"," 5),(0",", "otherwise"):}` If a = P(X ≤ 2) and b = P(X ≥ 3), then

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 ______

If a d.r.v. X has the following probability distribution:

X |
1 | 2 | 3 | 4 | 5 | 6 | 7 |

P(X = x) |
k | 2k | 2k | 3k | k^{2} |
2k^{2} |
7k^{2} + k |

then k = ______

Find mean for the following probability distribution.

X |
0 | 1 | 2 | 3 |

P(X = x) |
`1/6` | `1/3` | `1/3` | `1/6` |

Find the expected value and variance of r.v. X whose p.m.f. is given below.

X |
1 | 2 | 3 |

P(X = x) |
`1/5` | `2/5` | `2/5` |

**The probability distribution of X is as follows:**

X |
0 | 1 | 2 | 3 | 4 |

P(X = x) |
0.1 | k | 2k | 2k | k |

Find k and P[X < 2]

**Choose the correct alternative:**

f(x) is c.d.f. of discete r.v. X whose distribution is

x_{i} |
– 2 | – 1 | 0 | 1 | 2 |

p_{i} |
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) |
q^{2} |
2pq | p^{2} |

then Var(x) = ______

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

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 |

Complete the following activity.

**Solution:** Since `sum"p"_"i"` = 1

P(X ≥ 3) = `square - square - square = square`

Using the following activity, find the expected value and variance of the r.v.X if its probability distribution is as follows.

x |
1 | 2 | 3 |

P(X = x) |
`1/5` | `2/5` | `2/5` |

**Solution:** µ = E(X) = `sum_("i" = 1)^3 x_"i""p"_"i"`

E(X) = `square + square + square = square`

Var(X) = `"E"("X"^2) - {"E"("X")}^2`

= `sum"X"_"i"^2"P"_"i" - [sum"X"_"i""P"_"i"]^2`

= `square - square`

= `square`

The following function represents the p.d.f of a.r.v. X

f(x) = `{{:((kx;, "for" 0 < x < 2, "then the value of K is ")),((0;, "otherwise")):}` ______

**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)

**The probability distribution of X is as follows:**

x |
0 | 1 | 2 | 3 | 4 |

P[X = x] |
0.1 | k | 2k | 2k | k |

Find

- k
- P[X < 2]
- P[X ≥ 3]
- P[1 ≤ X < 4]
- P(2)

**The p.m.f. of a random variable X is as follows:**

P (X = 0) = 5k^{2}, P(X = 1) = 1 – 4k, P(X = 2) = 1 – 2k and P(X = x) = 0 for any other value of X. Find k.

Given below is the probability distribution of a discrete random variable x.

X |
1 | 2 | 3 | 4 | 5 | 6 |

P(X = x) |
K | 0 | 2K | 5K | K | 3K |

Find K and hence find P(2 ≤ x ≤ 3)