Question #184187

II. Find the expected value, variance, and standard deviation of the following discrete 

probability distributions. (3 items x 15 points) 

 

1.

T -3 5 7 

P(T) 0.27 0.4 0.33 

 

2.

X 0 1 2 3 4 

P(x=X) 0.21 0.44 0.06 0.11 0.18 

3.

Y 27 30 14 43 

P(Y=y) 0.3 0.4 0.15 0.15


1
Expert's answer
2021-05-07T08:55:59-0400

1.

T357P(T=t)0.270.40.33\begin{array}{|c|c|c|c|}\hline T& -3& 5& 7\\ \hline P(T=t)& 0.27& 0.4& 0.33\\ \hline\end{array}


Check the sum of probabilities

0.27+0.4+0.33=10.27+ 0.4+ 0.33=1


The expected value is

E[T]=30.27+50.4+70.33=3.5\bold{E}[T]= -3\cdot0.27+ 5\cdot0.4+ 7\cdot0.33 =3.5


The variance is

Var[T]=E[T2]E2[T]=\bold{Var}[T]=\bold{E}[T^2]-\bold{E}^2[T]= ((3)20.27+520.4+720.33)(3.5)2=\bigg((-3)^2\cdot0.27+ 5^2\cdot0.4+ 7^2\cdot0.33\bigg)-(3.5)^2 = 16.3516.35


The standard deviation is

σ[T]=Var[T]=16.354.04\sigma[T]=\sqrt{\bold{Var}[T]}=\sqrt{16.35}\approx4.04



2.

X01234P(X=x)0.210.440.060.110.18\begin{array}{|c|c|c|c|c|c|}\hline X &0& 1& 2& 3& 4 \\ \hline P(X=x) &0.21& 0.44& 0.06& 0.11& 0.18\\ \hline\end{array}


Check the sum of probabilities

0.21+0.44+0.06+0.11+0.18=10.21+ 0.44 +0.06+ 0.11+ 0.18=1


The expected value is

E[X]=\bold{E}[X]= 00.21+10.44+20.06+30.11+40.18=0\cdot0.21+1\cdot0.44+2\cdot0.06+3\cdot0.11+4\cdot0.18= 1.611.61


The variance is

Var[X]=\bold{Var}[X]= (020.21+120.44+220.06+320.11+420.18)(1.61)21.96(0^2\cdot0.21+1^2\cdot0.44+2^2\cdot0.06+3^2\cdot0.11+4^2\cdot0.18)-(1.61)^2\approx1.96


The standard deviation is

σ[X]=1.961.40\sigma[X]=\sqrt{1.96}\approx1.40



3.

Y27301443P(Y=y)0.30.40.150.15\begin{array}{|c|c|c|c|c|c|}\hline Y& 27& 30& 14& 43 \\ \hline P(Y=y)& 0.3& 0.4& 0.15& 0.15\\ \hline\end{array}


Check the sum of probabilities

0.3+0.4+0.15+0.15=10.3 +0.4+ 0.15+ 0.15=1


The expected value is

E[Y]=270.3+300.4+140.15+430.15=\bold{E}[Y]=27\cdot0.3+ 30\cdot0.4 + 14\cdot0.15 + 43\cdot0.15= 28.6528.65


The variance is

Var[Y]=\bold{Var}[Y]= (2720.3+3020.4+1420.15+4320.15)(28.65)264.63(27^2\cdot0.3+ 30^2\cdot0.4 + 14^2\cdot0.15 + 43^2\cdot0.15)-(28.65)^2\approx64.63


The standard deviation is

σ[Y]=64.638.04\sigma[Y]=\sqrt{64.63}\approx8.04



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