The discrete random variable X has positive probability on the nonnegative integ

Question

The discrete random variable X has positive probability on the nonnegative integers. The distribution has tail probabilities P (X > 2) 1/(z + 1) for x = 0, 1, 2, . . . . (a) Evaluate numerically P(X > 0), P(X > 1), and P(X = 0). (b) The probability P (X-can be expressed as a fraction of the form TT. m wtaTE for sone numbers a for some numbers a and b. Find P(X = r) in this form. (c) Using R (or another program), plot the probability mass function for X. Using R, plot Eo P(x 2j) as a function of k. What do you notice as k gets large?

Explanation / Answer

a. P(X>=x)=1/(x+1) , x=0,1,2,.......

P(X>=0)=1

P(X>=1)=1/2

P(X=0)= P(X>=0)-P(X>=1)=1-1/2=1/2

b. P(X=x)=P(X>=x)-P(X>=x+1)=1/(x+1)-1/(x+2)=1/(x+1)(x+2), here a=1,b=2

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