#### Random Variable and Mathematical Expectation Model Questions

12th Standard EM

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Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. A variable that can assume any possible value between two points is called

(a)

discrete random variable

(b)

continuous random variable

(c)

discrete sample space

(d)

random variable

2. A discrete probability distribution may be represented by

(a)

table

(b)

graph

(c)

mathematical equation

(d)

all of these

3. If we have f(x)=2x, 0$\le$x$\le$1, then f (x) is a

(a)

probability distribution

(b)

probability density function

(c)

distribution function

(d)

continuous random variable

4. A set of numerical values assigned to a sample space is called

(a)

random sample

(b)

random variable

(c)

random numbers

(d)

random experiment

5. The distribution function F(x) is equal to

(a)

P(X-x)

(b)

P(X$\le$x)

(c)

P(X$\ge$x)

(d)

all of these

6. 6 x 2 = 12
7. The discrete random variable X has the following probability function
P(X=x)={$\\ \\ kx\quad \quad \quad \quad x=2,4,6\\ k(x-2)\quad x=8\\ 0\quad \quad\quad \quad otherwisde$
where k is a constant. Show that k=$\frac{1}{18}$

8. A continuous random variable X has the following distribution function:
F(x)={$0\quad \quad \quad \quad ,if\quad x\le 1\\ k(x-1{ ) }^{ 4 },if13$
Find (i) k and (ii) the probability density function.

9. Distinguish between discrete and continuous random variable.

10. A person tosses a coin and is to receive Rs. 4 for a head and is to pay Rs. 2 for a tail. Find the expectation and variance of his gains.

11. 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 (a) k and (b) P(X>2).

12. The probability function of a random variable X is given by
p(x)={$\frac { 1 }{ 4 } ,\quad for\quad x=-2\\ \frac { 1 }{ 4 } ,\quad for\quad x=0\\ \frac { 1 }{ 2 } ,\quad for\quad x=10\\ 0,\quad elsewhere$
Evaluate the following probabilities.
P(|X|$\le$2)

13. 6 x 3 = 18
14. 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
15. A random variable X has the following probability function

 Values of X 0 1 2 3 4 5 6 7 p(x) 0 a 2a 2a 3a a2 2a2 7a2+a
16. A coin is tossed thrice. Let Xbe the number of observed heads. Find the cumulative distribution function of X.

17. A continuous random variable X has the following p.d.f
f(x)=ax, 0$\le$x$\le$1
Determine the constant a and also find P$\\ \left[ X\le \frac { 1 }{ 2 } \right]$

18. 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.

19. The amount of bread (in hundreds of pounds) x that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the probability density function f(x) is given  by
f(x)={$\\ \\ \\ Ax,\quad for\quad 0\le x10\\ A(20-x),\quad for\quad 10\le x<20\\ 0,\quad \quad \quad otherwise$
(a) Find the value of A.
(b) What is the probability that the number of pounds of bread that will be sold tomorrow is
(i) More than 10 pounds,
(ii) Less than 10 pounds, and
(iii) Between 5 and 15 pounds?

20. 3 x 5 = 15
21. An urn contains four balls of red, black, green and blue colours. There is an equal probability of getting any coloured ball. What is the expected value of getting a blue ball out of 30 experiments with replacement?

22. A commuter train arrives punctually at a station every 25 minutes. Each morning, a commuter leaves his house and casually walks to the train station. Let X denote the amount of time, in minutes, that commuter waits for the train from the time he reaches the train station. It is known that the probability density function of X is
f(x)={$\frac { 1 }{ 25 } ,\quad for0<x<25\\ 0,\quad otherwise,$
Obtain and interpret the expected value of the random variable X.

23. The probability density function of a random variable X is
f(x)=ke-|x|, -∞<x< ∞
Find the value of k and also find mean and variance for the random variable.