Question 2.
a) "X" is defined as the number of success (correct answers) in a series of independent experiments (answering the questions "at random") thus "X \\sim {\\text{Bin(}}n,p{\\text{)}}" - binomial distribution with "n = 100" and "p = \\frac{1}{5}" where probability mass function is
"{p_X}(k) = \\binom{n}{k}{p^k}{(1 - p)^{n-k}}=\\binom{100}{k}{(\\frac{1}{5})^k}{(\\frac{4}{5})^{100-k}}"where "\\binom{n}{k} = \\frac{n!}{k!(n-k)!}" - binomial coefficient
b) We need to find a formula for expected value "\\mathbb{E} X" . The easiest way to do it - represents our random variable "X" as a sum of other random variables (independent) "X = \\sum\\limits_{i = 1}^n {{Y_i}}" where "{{Y_i}}" is defined as a result of one experiment (yes-no). Its probability mass function is
"{p_Y}(k) = \\begin{cases} 1-p, & k=0 \\\\ p, & k=1 \\end{cases}"So it is easy to find its expected value
And now we can find the expected value of binomial distribution as (we use that expected value of sum is the sum of expected values)
Let's calculate it in our case
c) We can find the variance using the same way as in b). We shall use that
And now let's use that variance of sum is sum of variances in case of independent (or uncorrelated, more explicitly) random variables
Let's do the calculations
Standard deviation is the square root of variance
d) We need to calculate "P(X = 20) = {p_X}(20)=\\binom{100}{20}{p^{20}}{(1 - p)^{80}}" but it can't be done wihout computer. We can use De Moivre–Laplace theorem which claims "{\\text{Bin(}}n,p{\\text{)}} \\to N(np,np(1 - p))" as "n \\to + \\infty" and "p" remains fixed. So in means that
and now we can do the calculations
Question 3.
a) The probability mass function for Poisson distribution with mean "{\\lambda}" has the form
Thus probability that zero particles will be recorded is
b) We need to find the probability "P(X \\geqslant 4)" . We shall use that "\\sum\\limits_{k = 0}^n {\\frac{{{\\lambda ^k}{e^{ - \\lambda }}}}{{k!}}} = 1" thus
Do the calculations
Comments
Leave a comment