Give an answer with RStudio codes.

## Question 4 In a hamster breeder's experience, $X$ is the number of live pups in a litter of a female not over twelve months in age who has not borne a litter in the past six weeks with the probability distribution: *(10 pts.)* ***{r 44, echo=F} p4 ### Part a) Plot the probability mass function (pmf) of $X$ considering its range. Is this a valid probability mass function? *(1 pts.) ### solution of part a) {r 04 a} #write your codes here. Their results will appear after knitting the markdown file. ### Part b) obtain the cumulative distribution function (cdf) of $X$ considering its range. Plot the cdf. *(2 pts.) * ### Solution of part b) ***{r q4 b} #write your codes here. Their results will appear after knitting the markdown file. ### Part ) Find the probability that the next litter will produce 'r n41 **to** 'r n41+2 live pups. *(1 pts.) * || ### Solution of part c) ***{r 94 c} #write your codes here. Their results will appear after knitting the markdown file. ### Part d) Find the probability that the next litter will produce **at least** 'r n42 live pups. *(1 pts.)* ### Solution of part d) ***{r q4 d} #write your codes here. Their results will appear after knitting the markdown file. ### Part e Find the expected value, variance and standard deviation of $X$. * (2 pts. ) * ### solution of part e) ***{r 04 e} #write your codes here. Their results will appear after knitting the markdown file. ### Part f) Find the probability distribution of $y='s a4 X^2+'r b4 5. Plot the pmf of $y$ considering its range. Find the mean, variance and standard deviation of $y$. *(3 pts.) ### Solution of part f) ***{r 04 f} #Write your codes here. Their results will appear after knitting the markdown file. ## Question 4 In a hamster breeder's experience, $X$ is the number of live pups in a litter of a female not over twelve months in age who has not borne a litter in the past six weeks with the probability distribution: *(10 pts.)* ***{r 44, echo=F} p4 ### Part a) Plot the probability mass function (pmf) of $X$ considering its range. Is this a valid probability mass function? *(1 pts.) ### solution of part a) {r 04 a} #write your codes here. Their results will appear after knitting the markdown file. ### Part b) obtain the cumulative distribution function (cdf) of $X$ considering its range. Plot the cdf. *(2 pts.) * ### Solution of part b) ***{r q4 b} #write your codes here. Their results will appear after knitting the markdown file. ### Part ) Find the probability that the next litter will produce 'r n41 **to** 'r n41+2 live pups. *(1 pts.) * || ### Solution of part c) ***{r 94 c} #write your codes here. Their results will appear after knitting the markdown file. ### Part d) Find the probability that the next litter will produce **at least** 'r n42 live pups. *(1 pts.)* ### Solution of part d) ***{r q4 d} #write your codes here. Their results will appear after knitting the markdown file. ### Part e Find the expected value, variance and standard deviation of $X$. * (2 pts. ) * ### solution of part e) ***{r 04 e} #write your codes here. Their results will appear after knitting the markdown file. ### Part f) Find the probability distribution of $y='s a4 X^2+'r b4 5. Plot the pmf of $y$ considering its range. Find the mean, variance and standard deviation of $y$. *(3 pts.) ### Solution of part f) ***{r 04 f} #Write your codes here. Their results will appear after knitting the markdown file