Problem 22E

The morality of a solution in defined to be the number of moles of solute per liter of solution (1 mole=6.02×1023 molecules).if X is the morality of a solution of sodium choride(NaCl),and Y is the morality of a solution of sodium carbonate (Na2CO3) the morality of sodium ion (Na+),in a solution made of equal parts NaCl and Na2CO3 is given by M=0.5X+Y. Assume X and Y are independent and normally distributed, and that X has mean 0.450 and standard deviation 0.050,and Y has mean 0.025.

a. What is the distribution of M?

b. Find P(M >0.5)

Solution 22E

Step1 of 3:

We have random variable X it presents the morality of a solution of sodium chloride(NaCl) and another random variable Y presents the the morality of a solution of sodium carbonate

(). Let us assume that X and Y are independent and normally distributed, here X has mean and standard deviationand Y has meanand standard

deviation M is solution made of equal parts NaCl and That is

M = 0.5X + Y.

Here our goal is:

a).We need to find the distribution of M.

b).we need to find P(M > 0.5).

Step2 of 3:

a).

As per the given information we assume that X and Y are independent and normally distributed so, as per the property of normal distribution that is “Linear combination of independent and normally distributed variables is also normally distributed.”

Therefore, X and Y are independent and normally distributed hence M is also normally distributed.

That is MN(, )

Where,

1).= E(M)

Substitute M value in above equation we get

= E(0.5X+Y)

= 0.5E(X) + E(Y)

= 0.5+

= 0.5(0.450) + 0.250

= 0.225 + 0.250

= 0.475

Hence, = 0.475.

2).= Var(M)

Substitute M value in above equation we get

= Var(0.5X+Y)

= 0.5Var(X) + Var(Y)

=

=

= 0.25(0.0025) + 0.000625

= 0.000625 + 0.000625

= 0.00125

Hence, = 0.00125

3).

=

= 0.0353

Hence, = 0.0353.

Step3 of 3:

b).

Consider,

P(M > 0.5) = 1 -

= 1 -

= 1 -

= 1 -

Here is obtained from standard normal table(area under normal curve)

= 0.7580 (In statistical table we have to see row 0.7 under column 0.00)

Now,

= 1 -

= 1 - 0.7580

= 0.2420

Hence, P(M > 0.5) = 0.2420.