The probability mass function for X= number of major defects in a randomly selected appliance of a certain type is X =x01234P ( X =x)0.080.150.450.270..5 Find the expected value and variance of X.

E(X)= x_i . P (x_i ) =0(0.08)+1(0.15)+2(0.45)+3(0.27)+4(0.05) =0+0.15+0.9+0.81+0.2=2.06 E (X^2 )= x_i^2 P (x_i ) =0(0.08)+1^2(0.15)+2^2(0.45)+3^2(0.27)+4^2(0.05) =0+0.15+1.8+2.43+0.8=5.18 Var(X)=E (X^2 )-[E(X)]^2 =5.18-(2.06)^2 =5.18-4.2436 =0.9364

The probability mass function for X= number of major defects in a…

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Question

The probability mass function for $X=$ number of major defects in a randomly selected appliance of a certain type is

$X =x$	0	1	2	3	4
$P ( X =x)$	0.08	0.15	0.45	0.27	0..5

Find the expected value and variance of $X$.

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Answer

$E(X)=\sum x_i . P\left(x_i\right)$
$=0(0.08)+1(0.15)+2(0.45)+3(0.27)+4(0.05)$
$=0+0.15+0.9+0.81+0.2=2.06$
$E\left(X^2\right)=\sum x_i^2 \cdot P\left(x_i\right)$
$=0(0.08)+1^2(0.15)+2^2(0.45)+3^2(0.27)+4^2(0.05)$
$=0+0.15+1.8+2.43+0.8=5.18$
$\operatorname{Var}(X)=E\left(X^2\right)-[E(X)]^2$
$=5.18-(2.06)^2$
$=5.18-4.2436$
$=0.9364$

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