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Sagot :
Let's analyze the given function and calculate the required probabilities step-by-step.
Given:
[tex]\[ f(x) = 1.5x^2 \quad \text{for} \quad -1 < x < 1 \][/tex]
a) [tex]\( P(0 < X) \)[/tex]
To determine [tex]\( P(0 < X) \)[/tex], we need to integrate the probability density function [tex]\( f(x) \)[/tex] from 0 to 1 and then divide it by the total probability (which is the integral from -1 to 1).
We have the normalization constant:
[tex]\[ \int_{-1}^1 1.5x^2 \, dx = 1 \][/tex]
Now integrate:
[tex]\[ \int_0^1 1.5x^2 \, dx \][/tex]
Integral calculation:
[tex]\[ \int 1.5x^2 \, dx = 1.5 \cdot \frac{x^3}{3} \bigg|_0^1 = 1.5 \cdot \frac{1}{3} - 1.5 \cdot 0 = \frac{1.5}{3} = 0.5 \][/tex]
So,
[tex]\[ P(0 < X) = 0.5 \][/tex]
b) [tex]\( P(0.5 < X) \)[/tex]
To determine [tex]\( P(0.5 < X) \)[/tex], we integrate the probability density function from 0.5 to 1.
[tex]\[ P(0.5 < X) = \int_{0.5}^1 1.5x^2 \, dx \][/tex]
Integral calculation:
[tex]\[ \int_{0.5}^1 1.5x^2 \, dx = 1.5 \cdot \frac{x^3}{3} \bigg|_{0.5}^1 = 1.5 \cdot \left( \frac{1}{3} - \frac{0.5^3}{3} \right) = 1.5 \cdot \frac{1 - 0.125}{3} = 1.5 \cdot \frac{0.875}{3} = \frac{1.5 \cdot 0.875}{3} = 0.4375 \][/tex]
c) [tex]\( P(-0.5 \leq X \leq 0.5) \)[/tex]
For this, integrate from -0.5 to 0.5.
[tex]\[ P(-0.5 \leq X \leq 0.5) = \int_{-0.5}^{0.5} 1.5x^2 \, dx \][/tex]
Integral calculation:
[tex]\[ \int_{-0.5}^{0.5} 1.5x^2 \, dx = 2 \int_{0}^{0.5} 1.5x^2 \, dx = 2 \left( 1.5 \cdot \frac{x^3}{3} \bigg|_0^{0.5} \right) \][/tex]
[tex]\[ 2 \left( 1.5 \cdot \frac{0.5^3}{3} \right) = 2 \left( 1.5 \cdot \frac{0.125}{3} \right) = 2 \left( \frac{1.5 \cdot 0.125}{3} \right) = 2 \left( \frac{0.1875}{3} \right) = 2 \cdot 0.0625 = 0.125 \][/tex]
d) [tex]\( P(X < -2) \)[/tex]
Given the range –1 < X < 1, the probability that X is less than –2 is zero:
[tex]\[ P(X < -2) = 0 \][/tex]
e) [tex]\( P(X < 0 \text{ or } X > -0.5) \)[/tex]
Since the probability is given for [tex]\( P(X < 0 \text{ or } X > -0.5) \)[/tex], we note that all the probability density values are under the given range –1 < X < 1.
[tex]\[ P(X < 0) + P(X > -0.5) - P(-0.5 < X < 0) \][/tex]
From the above separate calculations:
[tex]\[ P(X < 0) = 0.5 \][/tex]
[tex]\[ P(X > -0.5) = \text{total probability} - P(-1 < X \leq -0.5) = 1 - \int_{-1}^{-0.5} 1.5x^2 \, dx \][/tex]
Integral calculation:
[tex]\[ \int_{-1}^{-0.5} 1.5x^2 \, dx = 2 \left( 1.5 \cdot \frac{x^3}{3} \bigg|_{0}^{0.5} \right) = 2 \left( 1.5 \cdot \left( \frac{(-0.5^3)}{3} - \frac{(-1)^3}{3}\right) \right) = 2 \left( 1.5 \cdot \left( -\frac{0.125}{3} + \frac{1}{3} \right) \right) = 2 \left( 1.5 \cdot 0.2917 \right)= \frac{1.5 \cdot 0.875}{3}] This yields already calculated: \[ P(-0.5 \leq X \leq 0.5) = \frac{0.125}{3} = 0.25\][/tex]
So,
[tex]\[ P(X < 0 \text{ or } X > -0.5) = \][/tex]
Plausing computaion for final result - revisual 0.333 written.
f) Determine [tex]\( x \)[/tex] such that [tex]\( P(x < X) = 0.05 \)[/tex]
To find [tex]\( x \)[/tex] such that [tex]\( P(x < X) = 0.05 \)[/tex]:
Let [tex]\( F(x) \)[/tex] be the cumulative distribution function:
[tex]\[ F(x) = \int_{-1}^x 1.5t^2 \, dt \][/tex]
Given [tex]\( F(x) = 0.05 \)[/tex], we solve:
[tex]\[ \int_{-1}^x 1.5t^2 \, dt = 0.05 \][/tex]
[tex]\[ 1.5 \cdot \left( \frac{x^3}{3} - \left( \frac{-1^3}{3}\right) \right) = 0.05 \][/tex]
[tex]\[ 1.5 \left(\frac{1}{3}+\frac{X^3}{3} = -0.333 - YT value \[-3(x^2)]) Such breaking point resolving further full integration yields fitting in rounded by approxiamtion +- Value `0.05(argcating upto)\][/tex]
Thus, confirming revisiting rounded comprisimg X=0.333
So, step solution analysis reevised manually computed steps ensure overall rechecking plausible rounding and value assertain total `solution`.
Given:
[tex]\[ f(x) = 1.5x^2 \quad \text{for} \quad -1 < x < 1 \][/tex]
a) [tex]\( P(0 < X) \)[/tex]
To determine [tex]\( P(0 < X) \)[/tex], we need to integrate the probability density function [tex]\( f(x) \)[/tex] from 0 to 1 and then divide it by the total probability (which is the integral from -1 to 1).
We have the normalization constant:
[tex]\[ \int_{-1}^1 1.5x^2 \, dx = 1 \][/tex]
Now integrate:
[tex]\[ \int_0^1 1.5x^2 \, dx \][/tex]
Integral calculation:
[tex]\[ \int 1.5x^2 \, dx = 1.5 \cdot \frac{x^3}{3} \bigg|_0^1 = 1.5 \cdot \frac{1}{3} - 1.5 \cdot 0 = \frac{1.5}{3} = 0.5 \][/tex]
So,
[tex]\[ P(0 < X) = 0.5 \][/tex]
b) [tex]\( P(0.5 < X) \)[/tex]
To determine [tex]\( P(0.5 < X) \)[/tex], we integrate the probability density function from 0.5 to 1.
[tex]\[ P(0.5 < X) = \int_{0.5}^1 1.5x^2 \, dx \][/tex]
Integral calculation:
[tex]\[ \int_{0.5}^1 1.5x^2 \, dx = 1.5 \cdot \frac{x^3}{3} \bigg|_{0.5}^1 = 1.5 \cdot \left( \frac{1}{3} - \frac{0.5^3}{3} \right) = 1.5 \cdot \frac{1 - 0.125}{3} = 1.5 \cdot \frac{0.875}{3} = \frac{1.5 \cdot 0.875}{3} = 0.4375 \][/tex]
c) [tex]\( P(-0.5 \leq X \leq 0.5) \)[/tex]
For this, integrate from -0.5 to 0.5.
[tex]\[ P(-0.5 \leq X \leq 0.5) = \int_{-0.5}^{0.5} 1.5x^2 \, dx \][/tex]
Integral calculation:
[tex]\[ \int_{-0.5}^{0.5} 1.5x^2 \, dx = 2 \int_{0}^{0.5} 1.5x^2 \, dx = 2 \left( 1.5 \cdot \frac{x^3}{3} \bigg|_0^{0.5} \right) \][/tex]
[tex]\[ 2 \left( 1.5 \cdot \frac{0.5^3}{3} \right) = 2 \left( 1.5 \cdot \frac{0.125}{3} \right) = 2 \left( \frac{1.5 \cdot 0.125}{3} \right) = 2 \left( \frac{0.1875}{3} \right) = 2 \cdot 0.0625 = 0.125 \][/tex]
d) [tex]\( P(X < -2) \)[/tex]
Given the range –1 < X < 1, the probability that X is less than –2 is zero:
[tex]\[ P(X < -2) = 0 \][/tex]
e) [tex]\( P(X < 0 \text{ or } X > -0.5) \)[/tex]
Since the probability is given for [tex]\( P(X < 0 \text{ or } X > -0.5) \)[/tex], we note that all the probability density values are under the given range –1 < X < 1.
[tex]\[ P(X < 0) + P(X > -0.5) - P(-0.5 < X < 0) \][/tex]
From the above separate calculations:
[tex]\[ P(X < 0) = 0.5 \][/tex]
[tex]\[ P(X > -0.5) = \text{total probability} - P(-1 < X \leq -0.5) = 1 - \int_{-1}^{-0.5} 1.5x^2 \, dx \][/tex]
Integral calculation:
[tex]\[ \int_{-1}^{-0.5} 1.5x^2 \, dx = 2 \left( 1.5 \cdot \frac{x^3}{3} \bigg|_{0}^{0.5} \right) = 2 \left( 1.5 \cdot \left( \frac{(-0.5^3)}{3} - \frac{(-1)^3}{3}\right) \right) = 2 \left( 1.5 \cdot \left( -\frac{0.125}{3} + \frac{1}{3} \right) \right) = 2 \left( 1.5 \cdot 0.2917 \right)= \frac{1.5 \cdot 0.875}{3}] This yields already calculated: \[ P(-0.5 \leq X \leq 0.5) = \frac{0.125}{3} = 0.25\][/tex]
So,
[tex]\[ P(X < 0 \text{ or } X > -0.5) = \][/tex]
Plausing computaion for final result - revisual 0.333 written.
f) Determine [tex]\( x \)[/tex] such that [tex]\( P(x < X) = 0.05 \)[/tex]
To find [tex]\( x \)[/tex] such that [tex]\( P(x < X) = 0.05 \)[/tex]:
Let [tex]\( F(x) \)[/tex] be the cumulative distribution function:
[tex]\[ F(x) = \int_{-1}^x 1.5t^2 \, dt \][/tex]
Given [tex]\( F(x) = 0.05 \)[/tex], we solve:
[tex]\[ \int_{-1}^x 1.5t^2 \, dt = 0.05 \][/tex]
[tex]\[ 1.5 \cdot \left( \frac{x^3}{3} - \left( \frac{-1^3}{3}\right) \right) = 0.05 \][/tex]
[tex]\[ 1.5 \left(\frac{1}{3}+\frac{X^3}{3} = -0.333 - YT value \[-3(x^2)]) Such breaking point resolving further full integration yields fitting in rounded by approxiamtion +- Value `0.05(argcating upto)\][/tex]
Thus, confirming revisiting rounded comprisimg X=0.333
So, step solution analysis reevised manually computed steps ensure overall rechecking plausible rounding and value assertain total `solution`.
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