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

To find the standard deviation, we first need to calculate the variance Var(X).

1. Calculate E(X²):
   $E(X²) = \sum_{x} x^2 \cdot P(X=x)$
   $= 0^2 \cdot 0 + 1^2 \cdot 0.3 + 2^2 \cdot 0.5 + 3^2 \cdot 0.1 + 4^2 \cdot 0.1$
   $= 0 + 0.3 + 2 + 0.9 + 1.6$
   $= 4.8$

2. Now find the variance:
   $Var(X) = E(X²) - [E(X)]²$
   $= 4.8 - 2^2$
   $= 4.8 - 4$
   $= 0.8$

3. Finally, calculate the standard deviation:
   $\sigma = \sqrt{Var(X)} = \sqrt{0.8} \approx 0.8944$

So the standard deviation, correct to one decimal place, is 0.9.