HSC Commerce Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions for Business Mathematics and Statistics

Prove that  Δ∇ = Δ – ∇

Prove that EV = Δ = ∇E

Construct cumulative distribution function for the given probability distribution.

Let X be a discrete random variable with the following p.m.f
`"P"(x) = {{:(0.3,  "for"  x = 3),(0.2,  "for"  x = 5),(0.3,  "for"  x = 8),(0.2,  "for"  x = 10),(0,  "otherwise"):}`
Find and plot the c.d.f. of X.

The discrete random variable X has the probability function

Show that k = 0 1

The discrete random variable X has the following probability function.
P(X = x) = `{{:("k"x,  x = 2","  4","  6),("k"(x - 2),  x = 8),(0,  "otherwise"):}`
where k is a constant. Show that k = `1/18`

Two coins are tossed simultaneously. Getting a head is termed a success. Find the probability distribution of the number of successes

The discrete random variable X has the probability function.

Find k

The discrete random variable X has the probability function.

Evaluate p(x < 6), p(x ≥ 6) and p(0 < x < 5)

The discrete random variable X has the probability function.

If P(X ≤ x) > `1/2`, then find the minimum value of x.

The distribution of a continuous random variable X in range (– 3, 3) is given by p.d.f.
f(x) = `{{:(1/16(3 + x)^2",", - 3 ≤ x ≤ - 1),(1/16(6 - 2x^2)",", - 1 ≤ x ≤ 1),(1/16(3 - x)^2",", 1 ≤ x ≤ 3):}`
Verify that the area under the curve is unity.

A continuous random variable X has the following distribution function
F(x) = `{{:(0",",  "if"  x ≤ 1),("k"(x - 1)^4",",  "if"  1 < x ≤ 3),(1",",  "if"  x > 3):}`
Find k

A continuous random variable X has the following distribution function
F(x) = `{{:(0",",  "if"  x ≤ 1),("k"(x - 1)^4",",  "if"  1 < x ≤ 3),(1",",  "if"  x > 3):}`
Find the Probability density function

The length of time (in minutes) that a certain person speaks on the telephone is found to be random phenomenon, with a probability function specified by the probability density function f(x) as 
f(x) = `{{:("Ae"^((-x)/5)",",  "for"  x ≥ 0),(0",",  "otherwise"):}`
Find the value of A that makes f(x) a p.d.f.

The length of time (in minutes) that a certain person speaks on the telephone is found to be random phenomenon, with a probability function specified by the probability density function f(x) as 
f(x) = `{{:("Ae"^((-x)/5)",",  "for"  x ≥ 0),(0",",  "otherwise"):}`
What is the probability that the number of minutes that person will talk over the phone is (i) more than 10 minutes, (ii) less than 5 minutes and (iii) between 5 and 10 minutes

Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function

F(x) = `{{:(0",",  "for"  x ≤ 0),(x/2",",  "for"  0 ≤ x < 1),(1/2",",  "for" ≤ x < 2),(x/4",",  "for"  2 ≤ x < 4),(1",",  "for"  x ≥ 4):}` 
Is the distribution function continuous? If so, give its probability density function?

Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function

F(x) = `{{:(0",",  "for"  x ≤ 0),(x/2",",  "for"  0 ≤ x < 1),(1/2",",  "for" ≤ x < 2),(x/4",",  "for"  2 ≤ x < 4),(1",",  "for"  x ≥ 4):}` 
What is the probability that a person will have to wait (i) more than 3 minutes, (ii) less than 3 minutes and (iii) between 1 and 3 minutes?

Define random variable

Explain what are the types of random variable?

Define dicrete random Variable