###### Advertisements

###### Advertisements

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)

###### Advertisements

#### Solution

We have `sumP(X = x)` = 1

∴ K + 0 + 2K + 5K + K + 3K = 1

∴ 12K = 1,

∴ K = `1/12`

Hence the probability distribution is

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

P(X = x) |
`1/12` | 0 | `2/12` | `5/12` | `1/12` | `3/12` |

∴ P(2 ≤ x ≤ 3) = P(x = 2) + P(x = 3)

= `0 + 2/12`

= `1/6`

#### 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

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.

Y |
−1 | 0 | 1 |

P(Y) |
0.6 | 0.1 | 0.2 |

X | 0 | -1 | -2 |

P(X) | 0.3 | 0.4 | 0.3 |

**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 expected value and variance of X for the following p.m.f.

x |
-2 | -1 | 0 | 1 | 2 |

P(X) |
0.2 | 0.3 | 0.1 | 0.15 | 0.25 |

The following is the p.d.f. of r.v. X:

f(x) = `x/8`, for 0 < x < 4 and = 0 otherwise.

Find P (x < 1·5)

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)

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)`.

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

P.d.f. of a.c.r.v X is f (x) = 6x (1 − x), for 0 ≤ x ≤ 1 and = 0, otherwise (elsewhere)

If P (X < a) = P (X > a), then a =

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

If the p.d.f of a.c.r.v. X is f (x) = x`^2/ 18` , for −3 < x < 3 and = 0, otherwise then P (| X | < 1) =

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

If a d.r.v. X takes values 0, 1, 2, 3, . . . which probability P (X = x) = k (x + 1)·5 ^{−x} , where k is a constant, then P (X = 0) =

**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 = x) = `x^2 /(n (n + 1))`, for x = 1, 2, 3, . . ., n and = 0, otherwise 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) =

Choose the correct option from the given alternative:

Find expected value of and variance of X for the following p.m.f.

X |
-2 | -1 | 0 | 1 | 2 |

P(x) |
0.3 | 0.3 | 0.1 | 0.05 | 0.25 |

**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

Find the probability distribution of number of heads in four tosses of a coin

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).

F(x) is c.d.f. of discrete 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) = _______ .

**Fill in the blank :**

If X is discrete random variable takes the value x_{1}, x_{2}, x_{3},…, xn then \[\sum\limits_{i=1}^{n}\text{P}(x_i)\] = _______

**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

**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 |

Determine the value of k.

**Solve the following problem :**

Let the p. m. f. of the r. v. X be

`"P"(x) = {((3 - x)/(10)", ","for" x = -1", "0", "1", "2.),(0,"otherwise".):}`

Calculate E(X) and Var(X).

**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 takes values 0, 1, 2, 3, … with probability P(X = x) = k(x + 1) × 5^{–x}, where k is a constant, then P(X = 0) = ______

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 the p.m.f. of a d.r.v. X is P(X = x) = `{{:(x/("n"("n" + 1))",", "for" x = 1"," 2"," 3"," .... "," "n"),(0",", "otherwise"):}`, then E(X) = ______

If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(("c")/x^3",", "for" x = 1"," 2"," 3","),(0",", "otherwise"):}` then E(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 ______

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 = ______

**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]

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than 4 appears on at least one die.

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) = ______

The values of discrete r.v. are generally obtained by ______

If X is discrete random variable takes the values x_{1}, x_{2}, x_{3}, … x_{n}, then `sum_("i" = 1)^"n" "P"(x_"i")` = ______

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`

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 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) = _______.