HSC Commerce इयत्ता १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions for Business Mathematics and Statistics

The probability function of a random variable X is given by
p(x) = `{{:(1/4",",  "for"  x = - 2),(1/4",",  "for"  x = 0),(1/2",",  "for"  x = 10),(0",",  "elsewhere"):}`
Evaluate the following probabilities
P(X < 0)

The probability function of a random variable X is given by
p(x) = `{{:(1/4",",  "for"  x = - 2),(1/4",",  "for"  x = 0),(1/2",",  "for"  x = 10),(0",",  "elsewhere"):}`
Evaluate the following probabilities
P(|X| ≤ 2)

The probability function of a random variable X is given by
p(x) = `{{:(1/4",",  "for"  x = - 2),(1/4",",  "for"  x = 0),(1/2",",  "for"  x = 10),(0",",  "elsewhere"):}`
Evaluate the following probabilities
P(0 ≤ X ≤ 10)

Let X be a random variable with a cumulative distribution function.
F(x) = `{{:(0",",  "if"  x  < 0),(x/8",",  "if"  0 ≤ x ≤ 1),(1/4 + x/8",",  "if"  1 ≤ x ≤ 2),(3/4 + x/12",",  "if"  2 ≤ x < 3),(1",",  "for"  3 ≤ x):}`
Compute: (i) P(1 ≤ X ≤ 2) and (ii) P(X = 3)

Let X be a random variable with a cumulative distribution function.
F(x) = `{{:(0",",  "if"  x  < 0),(x/8",",  "if"  0 ≤ x ≤ 1),(1/4 + x/8",",  "if"  1 ≤ x ≤ 2),(3/4 + x/12",",  "if"  2 ≤ x < 3),(1",",  "for"  3 ≤ x):}`
Is X a discrete random variable? Justify your answer

The p.d.f. of X is defined as
f(x) = `{{:("k"",",  "for"  0 < x ≤ 4),(0",",  "otherwise"):}`
Find the value of k and also find P(2 ≤ X ≤ 4)

The probability distribution function of a discrete random variable X is
f(x) = `{{:(2k",",  x = 1),(3k",",  x = 3),(4k",", x = 5),(0",",  "otherwise"):}`
where k is some constant. Find k 

The probability distribution function of a discrete random variable X is
f(x) = `{{:(2k",",  x = 1),(3k",",  x = 3),(4k",", x = 5),(0",",  "otherwise"):}`
where k is some constant. Find P(X > 2) 

The probability density function of a continuous random variable X is
f(x) = `{{:("a" + "b"x^2",",  0 ≤ x ≤ 1),(0",",  "otherwise"):}`
where a and b are some constants. Find a and b if E(X) = `3/5`

The probability density function of a continuous random variable X is
f(x) = `{{:(a + bx^2",",  0 ≤ x ≤ 1),(0",",  "otherwise"):}`
where a and b are some constants. Find Var(X)

Consider a random variable X with p.d.f.
f(x) = `{(3x^2",",  "if"  0 < x < 1),(0",",  "otherwise"):}`
Find E(X) and V(3X – 2)

Define Binomial distribution

Define Bernoulli trials

Derive the mean and variance of binomial distribution

Write down the condition for which the binomial distribution can be used.

Mention the properties of binomial distribution.

If 5% of the items produced turn out to be defective, then find out the probability that out of 20 items selected at random there are exactly three defectives

If 5% of the items produced turn out to be defective, then find out the probability that out of 20 items selected at random there are atleast two defectives

If 5% of the items produced turn out to be defective, then find out the probability that out of 20 items selected at random there are exactly 4 defectives

If 5% of the items produced turn out to be defective, then find out the probability that out of 20 items selected at random there are find the mean and variance