A discrete random variable X has a probability distribution function defined as \$P(X=x) = kx^2\$
where: x={0, 1, 2, 3,4} Calculate the probability that X=2.?

Sum of the probabilities is one

Finding the value of k

Substitute x={0, 1, 2, 3,4} in \$P(X=x) = kx^2\$

Sum of the probabilites = 1

Finding the value of k

\$\sum_{i=0}^4 P(X) = 1\$

Expand \$\sum_{i=0}^4 P(X)\$

P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)

Substituting the values in \$P(X=x) = kx^2\$

\$k0^2 + k1^2 + k2^2 + k3^2 + k4^2 = 1\$

Simplification

\$0 + k + 4k + 9k + 16k = 1\$

Value of K

\$30k = 1\$ And then \$k = 1/30\$

P(X=x) = \$\sum_{i=1}^n x_ip_i\$

Finding the P(X=2)

P(X=2) = \$(1/30) × 2^2\$

Simplification

P(X=2) = 0.03 × 4

After simplification

P(X=2) = 0.133

0 < \$p_i\$ < 1 Condition is satisfied

Answer

D