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Worked solution

Complete a distribution and read inequalities

The discrete random variable YY has probability distribution

yy−3-3−2-2−1-101
P(Y=y)P(Y=y)0.10.3dd0.250.01

Find (i) dd, (ii) P(−3≤Y<0)P(-3\le Y<0) and (iii) P(Y>−1)P(Y>-1).

CJC · Lecture · Example 2

(i) Normalize the five probabilities

Since ∑yP(Y=y)=1\sum_yP(Y=y)=1, 0.1+0.3+d+0.25+0.01=10.1+0.3+d+0.25+0.01=1, so d=0.34d=0.34.

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(ii) Include −3 and exclude zero

P(−3≤Y<0)=P(Y=−3)+P(Y=−2)+P(Y=−1)=0.1+0.3+0.34=0.74P(-3\le Y<0)=P(Y=-3)+P(Y=-2)+P(Y=-1)=0.1+0.3+0.34=0.74.

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(iii) Take only zero and one

P(Y>−1)=P(Y=0)+P(Y=1)=0.25+0.01=0.26P(Y>-1)=P(Y=0)+P(Y=1)=0.25+0.01=0.26.

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Common mistakes

A strict boundary excludes the boundary value even in a discrete distribution. List the permitted values before adding their probabilities.

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