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(Solved): 1. Let X be a continuous random variable with probability density function f(x)={32xfor1< ...

$1. Let \( \mathrm{X} \) be a continuous random variable with probability density function \[ f(x)=\left\{\begin{array}{c} \fr$

1. Let

X

be a continuous random variable with probability density function

f (x) = {\frac{2}{3} x for 1 < x < 2 0 elsewhere

Find

P (X \leq 1.2), P (X > 1.2), P (1.2 \leq X \leq 1.6), P (X = 1.5), P (X \leq .2)

, and

P (X > 3)

.

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Expert Answer

1.

   To find this probability, we need to integrate the PDF from negative 1 to 1.2:
P(X ? 1.2) =
Evaluating the integral, we get:


Therefore,

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