12th Business Maths Weekly Test - 5 ( Random Variable and Mathematical expectation )

St. Britto Hr. Sec. School - Madurai

12th Business Maths Weekly Test - 5 ( Random Variable and Mathematical expectation )-Aug 2020

Time : 1;00 Hours

Part - A

Generic 2 - Answer All Question

Prove that, (i) V(aX)=a²V(X), and
ii) V(X+b)=V(X)

How do you define variance in terms of Mathematical expectation?

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)=\(0,\quad for\quad x\le 0\\ \frac { x }{ 2 } ,\quad for\quad 0\le x<1\\ \frac { 1 }{ 2 } ,\quad for\quad 1\le x<2\\ \frac { x }{ 4 } ,\quad \ for\quad 2\le x<4\\ 1,\quad for\quad x\ge 4\)
(a) Is the distribution function continuous? If so, give its probability density function?
(b) 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?

Prove that,
(i) V(aX)=a²V(X), and
(ii) V(X+b) = V(X).

What do you understand by continuous random variable?

Let X be a continuous random variable with probability density function
f(x)={\(\frac { 3 }{ { x }^{ 4 } } ,\ge 1\\ 0,\quad otherwise\)
Find the mean and variance of X.

Part - B

Generic 3 - Answer All Question

Suppose, the life in hours of a radio tube has the following p.d.f
f(x)=\(\frac { 100 }{ { x }^{ 2 } } ,when\quad \ge 100\\ 0,\quad when\quad x<100\)
Find the distribution function.

The number of cars in a household is given below.

No. of cars 0 1 2 3 4

No. of Household 30 120 380 190 80

The probability distribution of a discrete random variable. X is given by

X -2 2 5

P (X = x) \(\frac14\) \(\frac14\) \(\frac12 \)

then find 4E(X²) - Var (2X)

A random variable. X has following distribution

X -1 0 1 2

P (X = x) \(\frac13\) \(\frac16\) \(\frac16\) \(\frac13\)

Find E(2X+3)²

A coin is tossed thrice. Let Xbe the number of observed heads. Find the cumulative distribution function of X.

A random variable X can take all nonnegative integral values and the probabilities that X takes the value r is proportional to a^r (0 < a < 1). Find P(X = 0).

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12th Business Maths Weekly Test - 5 ( Random Variable and Mathematical expectation )

St. Britto Hr. Sec. School - Madurai

12th Business Maths Weekly Test - 5 ( Random Variable and Mathematical expectation )-Aug 2020

Part - A

Generic 2 - Answer All Question

Prove that, (i) V(aX)=a2V(X), and ii) V(X+b)=V(X)

How do you define variance in terms of Mathematical expectation?

Prove that, (i) V(aX)=a2V(X), and (ii) V(X+b) = V(X).

What do you understand by continuous random variable?

Let X be a continuous random variable with probability density function f(x)={\(\frac { 3 }{ { x }^{ 4 } } ,\ge 1\\ 0,\quad otherwise\) Find the mean and variance of X.

Part - B

Generic 3 - Answer All Question

Suppose, the life in hours of a radio tube has the following p.d.f f(x)=\(\frac { 100 }{ { x }^{ 2 } } ,when\quad \ge 100\\ 0,\quad when\quad x<100\) Find the distribution function.

The number of cars in a household is given below. No. of cars 0 1 2 3 4 No. of Household 30 120 380 190 80

The probability distribution of a discrete random variable. X is given by X -2 2 5 P (X = x) \(\frac14\) \(\frac14\) \(\frac12 \) then find 4E(X2) - Var (2X)

A random variable. X has following distribution X -1 0 1 2 P (X = x) \(\frac13\) \(\frac16\) \(\frac16\) \(\frac13\) Find E(2X+3)2

A coin is tossed thrice. Let Xbe the number of observed heads. Find the cumulative distribution function of X.

A random variable X can take all nonnegative integral values and the probabilities that X takes the value r is proportional to ar (0 < a < 1). Find P(X = 0).

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Prove that, (i) V(aX)=a²V(X), and
ii) V(X+b)=V(X)

Prove that,
(i) V(aX)=a²V(X), and
(ii) V(X+b) = V(X).

Let X be a continuous random variable with probability density function
f(x)={\(\frac { 3 }{ { x }^{ 4 } } ,\ge 1\\ 0,\quad otherwise\)
Find the mean and variance of X.

Suppose, the life in hours of a radio tube has the following p.d.f
f(x)=\(\frac { 100 }{ { x }^{ 2 } } ,when\quad \ge 100\\ 0,\quad when\quad x<100\)
Find the distribution function.

The number of cars in a household is given below.

No. of cars 0 1 2 3 4

No. of Household 30 120 380 190 80

The probability distribution of a discrete random variable. X is given by

X -2 2 5

P (X = x) \(\frac14\) \(\frac14\) \(\frac12 \)

then find 4E(X²) - Var (2X)

A random variable. X has following distribution

X -1 0 1 2

P (X = x) \(\frac13\) \(\frac16\) \(\frac16\) \(\frac13\)

Find E(2X+3)²

A random variable X can take all nonnegative integral values and the probabilities that X takes the value r is proportional to a^r (0 < a < 1). Find P(X = 0).