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Questions
Find k, if the following function is p.d.f. of r.v.X:
f(x) = `{:(kx^2(1 - x)",", "for" 0 < x < 1),(0",", "otherwise"):}`
Find k, if f(x) = `{:(kx^2(1 - x)",", "for" 0 < x < 1),(0",", "otherwise"):}`
is the p.d.f. of random variable X.
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Solution 1
Since f(x) is p.d.f. of r.v.X, we get
`int_-∞^∞ f(x) dx` = 1
∴ `int_-∞^0 f(x)dx + int_0^1 f(x)dx + int_1^∞ f(x)dx` = 1
∴ `0 + int_0^1 kx^2 (1 - x)dx + 0` = 1
∴ `kint_0^1 (x^2 - x^3)dx` = 1
∴ `k[x^3/3 - x^4/4]_0^1` = 1
∴ `k[(1/3 - 1/4) - 0]` = 1
∴ `k(1/12)` = 1
∴ k = 12
Solution 2
Since (x) is the p.d.f. of a r.v. X,
∴ `int_0^1 kx^2 (1 - x) dx = 1`
∴ `int_0^1 k (x^2 - x^3)dx = 1`
`k int_0^1 (x^2 - x^3) dx = 1`
∴ `k{int_0^1 x^2 dx - int_0^1 x^3 dx} = 1`
∴ `k [x^3/3 - x^4/4]_0^1 = 1`
∴ `k(1/12) = 1`
∴ k = 12
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