The random variable $X$ has a probability distribution $P (X)$ of the following form, where $k$ is some number :
$P (X) = {k, i f x = 0 2 k, i f x = 1 3 k, i f x = 2 0, o t h e r w i s e}$
(a) Determine the value of $k$
(b) Find $P (X < 2), P (X \leq 2), P (X \geq 2)$ .

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Solution

It is known that the sum of probabilities of a probability distribution of random variables is one.
$∴ k + 2 k + 3 k + 0 = 1$
$\Rightarrow 6 k = 1$
$\Rightarrow k = \frac{1}{6}$
(b) $P (X < 2) = P (X = 0) + P (X = 1)$
$= k + 2 k$
$= 3 k$
$= \frac{3}{6}$
$= \frac{1}{2}$
$P (X \leq 2) = P (X = 0) + P (X = 1) + P (X = 2)$
$= k + 2 k + 3 k$
$= 6 k$
$= \frac{6}{6}$
$= 1$
$P (X \geq 2) = P (X = 2) + P (X > 2)$
$= 3 k + 0$
$= 3 k$
$= \frac{3}{6}$
$= \frac{1}{2}$

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The random variable X has a probability distribution P(X) of the following form, where k is some number :P(X)={k,ifx=02k,ifx=13k,ifx=20,otherwise}(a) Determine the value of k(b) Find P(X<2),P(X≤2),P(X≥2).

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$P (X) = {k, i f x = 0 2 k, i f x = 1 3 k, i f x = 2 0, o t h e r w i s e}$
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(b) Find $P (X < 2), P (X \leq 2), P (X \geq 2)$ .