Question

Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}`
where k is a constant. Calculate the value of k

Answer 1

Here, 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) = 2(5)k = 10k

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

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

And for otherwise it is 0.

∴ The probability distribution is given by

X	1	2	3	4	5	6	7	Otherwise
P(X)	2k	3k	4k	5k	10k	12k	14k	0

We know that `sum_("i" = 1)^"n" "P"("X"_"i")` = 1

So, 2k + 3k + 4k + 5k + 10k + 12k + 14k = 1

⇒ 50k = 1

⇒ k = `1/50`

Hence, the value of k is `1/50`