Let X be a discrete random variable
Question:

Let X be a discrete random variable whose probability distribution is defined as follows:

$P(\mathrm{X}=x)= \begin{cases}k(x+1) & \text { for } x=1,2,3,4 \\ 2 k x & \text { for } x=5,6,7 \\ 0 & \text { otherwise }\end{cases}$

where is a constant. Calculate

(i) the value of

(ii) E (X)

(iii) Standard deviation of X.

Solution:

(i) Given, P(X = x) = k(x + 1) for x = 1, 2, 3, 4

So, P(X = 1) = k(1 + 1) = 2k

P(X = 2) = k(2 + 1) = 3k

P(X = 3) = k(3 + 1) = 4k

P(X = 4) = k(4 + 1) = 5k

Also, P(X = x) = 2kx for x = 5, 6, 7

P(X = 5) = 2k(5) = 10k

P(X = 6) = 2k(6) = 12k

P(X = 7) = 2k(7) = 14k

And, for otherwise it is 0.

Thus, the probability distribution is given by