1. **Problem Statement:** A discrete random variable $X$ has a probability mass function (pmf) given by: $$f(x) = c(x+1), \quad x=0,1,2,3.$$ We need to find: (a) The constant $c$. (b) Draw the probability graph and histogram for $f(x)$. (c) Calculate probabilities: (i) $P(0 \leq X < 2)$, (ii) $P(X > 1)$. 2. **Finding the constant $c$:** Since $f(x)$ is a pmf, the sum of all probabilities must equal 1: $$\sum_{x=0}^3 f(x) = 1.$$ Substitute $f(x)$: $$c(0+1) + c(1+1) + c(2+1) + c(3+1) = 1,$$ which simplifies to: $$c(1 + 2 + 3 + 4) = 1,$$ $$c \times 10 = 1,$$ so $$c = \frac{1}{10} = 0.1.$$ 3. **Probability values:** Calculate $f(x)$ for each $x$: - $f(0) = 0.1 \times 1 = 0.1$ - $f(1) = 0.1 \times 2 = 0.2$ - $f(2) = 0.1 \times 3 = 0.3$ - $f(3) = 0.1 \times 4 = 0.4$ 4. **Probability graph and histogram:** - The probability graph is a discrete plot with points at $(0,0.1)$, $(1,0.2)$, $(2,0.3)$, $(3,0.4)$. - The probability histogram consists of bars at $x=0,1,2,3$ with heights $0.1, 0.2, 0.3, 0.4$ respectively. 5. **Calculating requested probabilities:** (i) $P(0 \leq X < 2) = P(X=0) + P(X=1) = 0.1 + 0.2 = 0.3.$ (ii) $P(X > 1) = P(X=2) + P(X=3) = 0.3 + 0.4 = 0.7.$ **Final answers:** - $c = 0.1$ - $P(0 \leq X < 2) = 0.3$ - $P(X > 1) = 0.7$