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#### Random Variable and Mathematical Expectation Two Marks Question

12th Standard EM

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Time : 00:45:00 Hrs
Total Marks : 30
15 x 2 = 30
1. 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=$\frac{1}{18}$

2. The discrete random variable X has the probability function

 X 1 2 3 4 P(X=x) k 2k 3k 4k

Show that k =0.1.

3. A continuous random variable X has the following distribution function:
$f(x)=\begin{cases} \begin{matrix} 0 & ifx\le 1 \end{matrix} \\ \begin{matrix} k(x-1)^{ 4 } & if1<x\le 3 \end{matrix} \\ \begin{matrix} 1, & ifx>3 \end{matrix} \end{cases}$
Find (i) k and (ii) the probability density function.

4. What do you understand by continuous random variable?

5. Describe what is meant by a random variable.

6. Distinguish between discrete and continuous random variable.

7. State the properties of distribution function.

8. Find the expected value for the random variable of an unbiased die

9. In an investment, a man can make a profit of Rs.5,000 with a probability of 0.62 or a loss of Rs. 8,000 with a probability of 0.38. Find the expected gain.

10. Define Mathematical expectation in terms of discrete random variable.

11. State the definition of Mathematical expectation using continuous random variable.

12. In a business venture a man can make a profit of Rs. 2,000 with a probability of 0.4 or have a loss of Rs.1,000 with a probability of 0.6. What is his expected, variance and standard deviation of profit?

13. The p.d.f. of X is defined as
$f(x)=\begin{cases} \begin{matrix} k & for0<x\le 4 \end{matrix} \\ \begin{matrix} 0, & otherwise \end{matrix} \end{cases}$
Find the value of k and also find P(2$\le$X$\le$4).

14. 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<0)

15. 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(0$\le$X$\le$10)