GATE 2020 ST – Question 30
Let $p(x,y,z)=P(X=x,Y=y,Z=z)$. The joint probability mass is $$\begin{aligned}p(x,y,z)&=\frac{10!}{x!y!z!k!}\,a,\\a&=(.2)^x(.3)^y(.4)^z(.1)^k.\end{aligned}$$ Here $k=10-x-y-z$, and x,y,z are nonnegative integers satisfying $x+y+z\le10$. Find Var(Y+Z), to one decimal place.
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Correct answer: 2.1 to 2.1
Explanation
Combining the Y and Z cells yields Y+Z∼Binomial(10,0.3+0.4)=Binomial(10,0.7). Its variance is np(1−p)=10(0.7)(0.3)=2.1. Combining categories preserves the total number of multinomial trials.