Question

A random variable X is defined by the function:
P(X)= 4/X!( 4-X)!(1/16) for x=0,1,2,3,4
Set up,in tabular form,the probability distribution for X.

Answer 1

$(i) Since f(x) is a p.d.f. ∫ - ∞ \: ∞ f(x)dx=1</p><p> \\ ⇒∫01k(1−x2)dx=1</p><p></p><p> \\ ⇒k[x−3x3]01=1</p><p> \\ ⇒k(1−31)=1</p><p></p><p> \\ ⇒32k=1</p><p></p><p> \\ ⇒k=23</p><p> \\ (ii) The distribution function F(x)=∫−∞xf(t)dt</p><p> \\ (a) When x∈(−∞,0]</p><p>F(x)=∫−∞xf(t)dt=0</p><p> \\ (b) When xε(0,1]</p><p>F(x)=∫−∞xf(t)dt=∫−∞0f(t)dt+∫0xf(t)dt</p><p> \\ =0+23∫0x(1−t2)dt</p><p>F(x)=23(x−3x3) \: \\ (c) When x∈[1,∞) \: \\ F(x)=∫−∞xf(t)dt</p><p> \\ =∫−∞0f(t)dt+∫01f(t)dt+∫1xf(t)dt</p><p> \\ =0+∫0123(1−t2)dt+0</p><p> \\ =23[t3t3]01=1</p><p> \\ ∴F(x)=⎩⎪⎪⎪⎨⎪⎪⎪⎧023(x−3x3),1−∞<x≤00<x<11≤x∞</p><p></p><p>$

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