Answer:
a
[tex]f(x)= 100 [x - 0.2050][/tex]
b
[tex]P(X > 0.2125) = 0.25 [/tex]
c
[tex]a = 0.2140[/tex]
d
Mean [tex]\mu = 0.21[/tex]
Variance [tex]\sigma ^2 = 0.0000083 [/tex]
Step-by-step explanation:
From the question we are told that
The thickness of photoresist follows a uniform distribution
So
[tex]X\ \~{}\ U \{ 0.2050 , 0.2150 \}[/tex]
Generally the probability density function is mathematically represented as
[tex]F(x) = \left \{ \frac{1}{ 0.2150 - 02050} , 0.2150< x < 0.2150[/tex]
=> [tex]F(x) = \left \{ 100 , 0.2150< x < 0.2150[/tex]
Generally the cumulative distribution function is
[tex]f(x) = \int\limits^{x}_{- \infty} {F(x)} \, dx[/tex]
Here [tex]- \infty[/tex] means the lower limit which in this case is 02050
So
[tex]f(x) = \int\limits^{x}_{0.2050} {100} \, dx[/tex]
[tex]f(x) = 100 \int\limits^{x}_{0.2050} {1} \, dx[/tex]
=> [tex] f(x)=100 [ x ] | \left \ x} \atop {0.2050}} \right.[/tex]
=> [tex]f(x)= 100 [x - 0.2050][/tex]
Generally the proportion of wafers that exceeds 0.2125 is mathematically represented as
[tex]P(X > 0.2125) = 1 - F(0.2125)[/tex]
=> [tex]P(X > 0.2125) = 1- 100[0.2125 - 0.2050][/tex]
=> [tex]P(X > 0.2125) = 0.25 [/tex]
Generally the thickness which is exceeded by 10% of the wafers is mathematically represented as
[tex]P(X > a ) = 0.10[/tex]
=> [tex]1 - F(a) = 0.10[/tex]
=> [tex]1 -100 (a - 0.2050) = 0.10[/tex]
=> [tex]100(a -0.2050) = 0.90[/tex]
=> [tex]a - 0.2050 = 0.009[/tex]
=> [tex]a = 0.2140[/tex]
Generally the mean is mathematically represented as
[tex]\mu = \frac{ 0.2050 -0.2150 }{2}[/tex]
=> [tex]\mu = 0.21[/tex]
Generally the variance is mathematically represented as
[tex]\sigma ^2 = \frac{1}{12} [0.2150 - 0.2050]^2[/tex]
=> [tex]\sigma ^2 = 0.0000083 [/tex]
