Probability
Probability distribution normalization
nta_pyq_2025_apr
Grade 12
Question:
If the probability that the random variable X takes the value x is given by P$(X = x) = k(x + 1)3 -x$,$x = 0$, 1, 2, 3$\ldots$$\ldots$, where k is a constant, then P(X$\ge$3) is equal to
$7 27$
$4 9$
$8 27$
$1 9$
Step-by-Step Solution
Key Concept: Compute the favourable cases for probability distribution normalization and divide by the total equally likely cases.
$\sum$$\infty$$x=0$$k(x + 1)3 -x = 1$(4) $\Rightarrow$ 1$k = 1 + 2$$3 + 3$$2 + 4$$3 + ...(i)$3 3 1$3k = 1$$3 + 2$$2 + 3$3 +$\ldots$...(ii) 3 3$(i)- (ii)$$\Rightarrow$ 1$k - 1$$3k = 1 + 1$$3 + 1$2 +$\ldots$3 4 $\Rightarrow$$k = 9$P(x$\ge$$3) = 1 - P(x = 0) - P(x = 1) - P(x = 2)$2 3$1 = 1 - k$(1 + +$) = 3$9 9
Correct Answer: 4