The sum of probabilities in a probability distribution must be one (i.e. 100%). Therefore, 0.2+0.3+x+0.1=1 and so x=0.1.

To find the expected value, E(X): \(1\times0.3+2\times0.4+3\times0.2+4\times0.1=2.1\)

Discrete random variables can only take a countable number of values. The marks on a test and the score on a die are both countable and so they are discrete random variables. In contrast, lengths of leaves and heights of students are measured, and they can take an infinite number of values. They are not discrete random variables.

The sum of probabilities in a probability distribution must be one (i.e. 100%). Therefore, \(k+2k+3k=1\) and so \(6k=1\), which means \(k=\frac{1}{6}\).

Given that we have two unknowns, we will create 2 simultneous equations to solve.

Equation 1: \(x+y+0.2+0.1=1\), and thus \(x+y=0.7\)

Equation 2: \(2\times x+3\times y+4 \times 0.1+5\times 0.2=3.4\), and thus \(2x+3y=2\)

Solving these linear simultaneous equations on the GDC, we obtain \(x=0.1\) and \(y=0.6\).

The expected value, \(E(X)=0\times0.3+1\times0.45+2\times0.2+3\times0.05=1 \).

If we substitute the given values of \(x \) into the function, we obtain the following table:

Since the probabilities in a probability distribution add up to one, then \(\frac{2}{k}+\frac{4}{k}+\frac{6}{k}=1\), so that \(\frac{12}{k}=1\) and hence \(k=12\).

\(P\left(2\le X\le3\right)\) is the probability that a student studies between 2 and 3 hours a night, including those values. That is 0.5 + 0.15 = 0.65.

Given that we have two unknowns, we will create 2 simultaneous equations to solve.

Equation 1: \(x+0.2+y+0.2=1\), and thus \(x+y=0.6\)

Equation 2: \(2\times x+3\times 0.2+4 \times y+5\times 0.2=3.8\), and thus \(2x+4y=2.2\)

Solving these linear simultaneous equations on the GDC, we obtain \(x=0.1\) and \(y=0.5\).

Given that we have two unknowns, we will create 2 simultaneous equations to solve.

Equation 1: \(0.3+x+y+0.2=1\), and thus \(x+y=0.5\)

Equation 2: \(2\times 0.3+3\times x+4 \times y+5\times 0.2=3.4\), and thus \(3x+4y=1.8\)

Solving these linear simultaneous equations on the GDC, we obtain \(x=0.2\) and \(y=0.3\).