SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 2.8 - Probability Distributions [Latest edition]

Choose the correct alternative:

The variance of a Binomial distribution is given by ______

Choose the correct alternative:

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

then F(– 3) = ______

X : is number obtained on upper most face when a fair die is thrown then E(X) = ______

If p.m.f. of r.v. X is given below.

then Var(x) = ______

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

If X ~ B`(20, 1/10)`, then E(X) = ______

Choose the correct alternative:

A sequence of dichotomous experiments is called a sequence of Bernoulli trials if it satisfies ______

Choose the correct alternative:

For the Poisson distribution ______

Choose the correct alternative:

A distance random variable X is said to have the Poisson distribution with parameter m if its p.m.f. is given by P(x) = `("e"^(-"m")"m"^"x")/("x"!)` the condition for m is ______

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

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

If X is discrete random variable takes the values x₁, x₂, x₃, … x_n, then `sum_("i" = 1)^"n" "P"(x_"i")` = ______

E(x) is considered to be ______ of the probability distribution of x.

In Binomial distribution, probability of success ______ from trial to trial

In Binomial distribution if n is very large and probability success of p is very small such that np = m (constant) then _______ distribution is applied.

When n is very large and p is very small in the binomial distribution, then X follows the Poission distribution with prameter m = ______

State whether the following statement is True or False:

X is the number obtained on upper most face when a die is thrown, then E(x) = 3.5

State whether the following statement is True or False:

If f(x) = `{:("k"x  (1 - x)",", "for"  0 < x < 1),(= 0",", "otherwise"):}`
is the p.d.f. of a r.v. X, then k = 12

State whether the following statement is True or False:

If X ~ B(n, p) and n = 6 and P(X = 4) = P(X = 2), then p = `1/2`

If r.v. X assumes the values 1, 2, 3, …….., 9 with equal probabilities, then E(X) = 5

State whether the following statement is True or False: 

Let X ~ B(n, p), then the mean or expected value of r.v. X is denoted by E(X). It is also denoted by E(X) and is given by µ = E(X) = npq

State whether the following statement is True or False:  

A discrete random variable X is said to follow the Poisson distribution with parameter m ≥ 0 if its p.m.f. is given by P(X = x) = `("e"^(-"m")"m"^"x")/"x"`, x = 0, 1, 2, .....

State whether the following statement is True or False:

For the Binomial distribution, Mean E(X) = m and Variance = Var(X) = m

State whether the following statement is True or False:

If n is very large and p is very small then X follows Poisson distribution with n = mp

State whether the following statement is True or False:

The cumulative distribution function (c.d.f.) of a continuous random variable X is denoted by F and defined by

F(x) = `{:(0",",  "for all"  x ≤ "a"),( int_"a"^x  f(x) "d"x",",  "for all"  x ≥ "a"):}`

Find the probability distribution of number of heads in two tosses of a coin.

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

A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs.

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

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

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes

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

If X has Poisson distribution with m = 1, then find P(X ≤ 1) given e⁻¹ = 0.3678

If X has Poisson distribution with parameter m and P(X = 2) = P(X = 3), then find P(X ≥ 2). Use e⁻³ = 0.0497

A random variable X has the following probability distribution:

Find:

1. k
2. P(X < 3)
3. P(X > 4)

The p.d.f. of a continuous r.v. X is

f(x) = `{((3x^2)/(8),  0 < x < 2),(0,   "otherwise".):}`
Determine the c.d.f. of X and hence find P(X < 1)

The p.d.f. of a continuous r.v. X is

f(x) = `{((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}`
Determine the c.d.f. of X and hence find P(X < –2)

The p.d.f. of a continuous r.v. X is

f(x) = `{((3x^2)/(8),  0 < x < 2),(0, "otherwise".):}`
Determine the c.d.f. of X and hence find P(X > 0)

The p.d.f. of a continuous r.v. X is

f(x) = `{((3x^2)/(8), 0 < x < 2),(0, "otherwise".):}`
Determine the c.d.f. of X and hence find P(1 < X < 2)

If a r.v. X has p.d.f f(x) = `{("c"/x","  1 < x < 3"," "c" > 0),(0","  "otherwise"):}` 
Find c, E(X), and Var(X). Also Find F(x).

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 successes

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at most 2 successes.

The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.

The number of complaints which a bank manager receives per day follows a Poisson distribution with parameter m = 4. Find the probability that the manager receives a) only two complaints on a given day, b) at most two complaints on a given day. Use e⁻⁴ = 0.0183.

Defects on plywood sheet occur at random with the average of one defect per 50 sq.ft. Find the probability that such a sheet has:

1. no defect
2. at least one defect
   Use e⁻¹ = 0.3678

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has exactly 5 rats inclusive. Given e^-5  = 0.0067.

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has more than 5 rats inclusive. Given e^-5  = 0.0067.

It is known that, in a certain area of a large city, the average number of rats per bungalow is five. Assuming that the number of rats follows Poisson distribution, find the probability that a randomly selected bungalow has between 5 and 7 rats inclusive. Given e⁻⁵ = 0.0067.

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

Complete the following activity.

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

k = `square`

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

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.

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.

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`

Let X ~ B(n, p). If n = 10 and E(X) = 5, using the following activity find p and Var(X)

Solution: E(X) = `square = 5 square "p" = square, "q" = square`

Var(X) = `square`

The probability that a bomb will hit the target is 0.8. Using the following activity, find the probability that, out of 5 bombs, exactly 2 will miss the target

Solution: Let p = probability that bomb miss the target

∴ q = `square`, p = `square`, n = 5.

X ~ B`(5, square)`, P(x) = `""^"n""C"_x"P"^x"q"^("n" - x)`

P(X = 2) =  `""^5"C"_2  square = square`

If X follows Poisson distribution such that P(X = 1) = 0.4 and P(X = 2) = 0.2, using the following activity find the value of m.

Solution: X : Follows Poisson distribution

∴ P(X) = `("e"^-"m" "m"^x)/(x!)`, P(X = 1) = 0.4 and P(X = 2) = 0.2

∴ P(X = 1) = `square` P(X = 2).

`("e"^-"m" "m"^x)/(1!) = square ("e"^-"m" "m"^2)/(2!)`,

`"e"^-"m" = square  "e"^-"m" "m"/2`, m ≠ 0

∴ m = `square`