The table shows the probability distribution of a discrete random variable - HSC - SSCE Mathematics Advanced - Question 12 - 2023 - Paper 1

To calculate the standard deviation, we first find the variance Var(X) using the following formula:

$Var(X) = E(X^2) - (E(X))^2$

We already found $E(X) = 2$. Now we need to compute $E(X^2)$:

• For $x = 0$: $0^2 imes P(X=0) = 0 imes 0 = 0$
• For $x = 1$: $1^2 imes P(X=1) = 1 imes 0.3 = 0.3$
• For $x = 2$: $2^2 imes P(X=2) = 4 imes 0.5 = 2.0$
• For $x = 3$: $3^2 imes P(X=3) = 9 imes 0.1 = 0.9$
• For $x = 4$: $4^2 imes P(X=4) = 16 imes 0.1 = 1.6$

Now, summing these values gives:

$E(X^2) = 0 + 0.3 + 2.0 + 0.9 + 1.6 = 5.8$

Now substituting back to find the variance:

$Var(X) = 5.8 - 2^2 = 5.8 - 4 = 1.8$

Finally, the standard deviation is the square root of the variance:

$ext{Standard deviation} = ext{sqrt}(1.8) \approx 1.3416$

Rounding to one decimal place gives:

$ext{Standard deviation} = 1.3$