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# Probability distribution of a random variable

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1.

Let the random variable X have the p.d.f.

f(x)=2(1-x) for 0<x<1 and zero elsewhere.

a. Sketch the graph
b. Determine and sketch the graph of the distribution function of X
c. find P(X) for the following intervals: i. [0,1/2] ii. [1/4,3/4] iii. X=3/4 iv. X>3/4

2.
For each of the following functions (i) find the constant c so that f(x) is a p.d.f of a random variable X, (ii) find the distribution function F(x)=P(X<x) and (iii) sketch graphs of the p.d.f f(x) and the distribution function F(x):

a. f(x)=(x^3)/4 0<x<c
b f(x)=3(x^2)/16 -c<x<c
c. f(x)= c/sqrt(x) 0<x<1

3.
For each of the following functions (i) find the constant c so that f(x) is a p.d.f of a random variable X, (ii) find the distribution function F(x)=P(X<x) and (iii) sketch graphs of the p.d.f f(x) and the distribution function F(x):

a. f(x)=4x^c 0<x<1
b. f(x)=c/sqrt(x) 0<x<4
c. c/(x^3/4) 0<x<1