Let X be a random variable with probability density function

$f (x) = {\begin{matrix} 0.2, f o r | x | \leq 1 \\ 0.1, f o r 1 < | x | \leq 4 \\ 0, o t h e r w i s e \end{matrix}$

The probability P(0.5 < X < 5) is _______.

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Answer 1

The given probability density function is

$f (x) = {\begin{matrix} 0.2, f o r | x | \leq 1 \\ 0.1, f o r 1 < | x | \leq 4 \\ 0, o t h e r w i s e \end{matrix}$

Now, $f (x) = {\begin{matrix} 0.2, x = (- 1, 1) \\ 0.1, x = (- 4, - 1) \cup (1, 4) \\ 0, o t h e r w i s e \end{matrix}$

The graph will be,

F1 Vinanti Engineering 20.01.23 D2

The probability P(0.5 < X < 5)

$(P) = \int_{- \infty}^{\infty} f (x) d x$

$= \int_{0.5}^{1} 0.2 d x + \int_{1}^{4} 0.1 d x$

= 0.2 [1 – 0.5] + 0.1 [4 - 1]

P = 0.2 × 0.5 + 0.1 × 3

= 0.1 + 0.3 = 0.4 (Shaded area)