p 2- p 3- p 4- p5- 1-Write a pdf for this distribution? 2-Derive the CDF express

Question

p 2-

p 3-

p 4-

p5-

1-Write a pdf for this distribution?

2-Derive the CDF expression for this distribution?

3-Find the mean of this distribution?

4-  Compute the variance of this distribution?

The actual volume of soup in 500ml jars follows a normal distribution with mean 500ml and variance 16ml. If X denotes the actual volume of soup in a jar, what is (i) POX 496)? (ii) P(X 498)? (iii) P(492 X 512)? (iv) P(X 480)? suppose that the mean volume of soup in a jar is unknown but that the standard deviation is 4. lf only 3% of jars are to contain less than 492ml what should the mean volume of soup in a jar be?

Explanation / Answer

1.
We are given params of Normal dist. We will standardize variable using them:
Mean = 500
Stdev =sqrt(variance) = 4

i)    P(X>496) = P(Z > -1) = .84
ii)   P(X<498) = P(Z < -.5) = .3085
iii) P(492<X<512) = P( -2<Z<2) = .95
iv) P(X>480) = P(Z> -5) = ~1

2nd part of p1)

Mean is unknown

Stdev =4
P(X<492) = .03
Z for .03 is -1.88
So, 492-Mu / 4 = -1.88

Mu = 492+4*-1.88 = 484.48

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