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)
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:
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(, )
Substitute M value in above equation we get
= 0.5E(X) + E(Y)
= 0.5(0.450) + 0.250
= 0.225 + 0.250
Hence, = 0.475.