The discrete random variable X has the following probability function \(P(X=x) = \begin{cases}kx & x =2, 4, 6 \\ k(x - 2), & x = 8 \\ 0, & \text { otherwise } \\ \end{cases}\) where k is a constant. Show that k = \(\frac{1}{18}\)

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.

A continuous random variable X has the following distribution function:
\(f(x)=\left\{\begin{array}{l} 0 , \text{if} \ x \leq1 \\ k(x-1)^4, \text{if} \ 1< x \leq 3 \\ 1, \text{if} \ x > 3 \end{array}\right.\)
Find (i) k and (ii) the probability density function.

What do you understand by continuous random variable?

Describe what is meant by a random variable.

Distinguish between discrete and continuous random variable.

State the properties of distribution function.

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

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.

Define Mathematical expectation in terms of discrete random variable.

State the definition of Mathematical expectation using continuous random variable.

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?

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

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
P(X<0)

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
 P(0\(\le\)X\(\le\)10)