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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability2 Marks
Question
The random variable X has a probability distribution P(X) of the following form, where k is some number :
$ \text{P}(\text{X}) = \begin{cases} \text{k, }\overline{\text{if}\ \text{x}=0} \\ \overline{ 2\text{k, }\text{if}\ \text{x}=1}\\3\text{k, }\text{if}\ \text{x}=2\\0,\ \text{otherwise} \end{cases}$
  1. Determine the value of k.
  2. Find $P(X < 2), P(X ≤ 2), P(X ≥ 2)$.
✓

Answer

Probability distribution:
$x_i$ 0 1 2
$P(x_i)$ k 2k 3k
  1. P(X = 0) + P(X = 1) + P(X = 2) = 1
⇒ k + 2k + 3k = 1 ⇒ 6k = 1
$\Rightarrow\ \text{k}=\frac{1}{6}$
  1. P(X < 2) = P(X = 0) + P(X = 1)
$=\text{k}+2\text{k}=3\text{k}=3\times\frac{1}{6}=\frac{1}{2}$
P(X ≥ 2) = P(X = 2) = 3k $=3\times\frac{1}{6}=\frac{1}{2}$

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