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प्रश्न
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
उत्तर
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`
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