Mean and Standard Deviation - AP Statistics

1. Identify the standard deviation of a Poisson random variable with rate $\text{lambda}$.

   Standard deviation is $\text{sqrt(lambda)}$. Poisson standard deviation is square root of rate parameter.

2. Calculate the variance for $P(X=1)=0.4$, $P(X=2)=0.6$. Mean is $1.6$.

   Variance is $0.24$. Sum $(x_i - 1.6)^2 \times p_i$ for each value.

3. Find the variance of a random variable $X$ with $Var(X)=9$ after $Y=2X-3$.

   $Var(Y) = 36$. Apply variance rule: $Var(Y) = 2^2 \times 9 = 36$.

4. What is the variance of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$?

   $Var(X) = \text{sigma}^2$. Normal distribution variance is $\sigma^2$ by definition.

5. What is the mean of a Poisson random variable with rate $\text{lambda}$?

   $E(X) = \text{lambda}$. For Poisson distribution, mean equals the rate parameter.

6. Calculate the mean of a random variable $X$ with $E(X) = 5$ after transformation $Y = 3X + 2$.

   $E(Y) = 17$. Apply linear transformation: $E(Y) = 3(5) + 2 = 17$.

7. State the formula for the variance of a discrete random variable $X$.

   $Var(X) = \text{sum of } [(x_i - \bar{x})^2 \times p_i]$. Each squared deviation from mean times its probability, then sum.

8. What is the effect on the variance if $X$ is shifted by $c$, i.e., $Y = X + c$?

   Variance remains unchanged, $Var(Y) = Var(X)$. Adding constant doesn't affect variability measures.

9. How do you calculate the standard deviation from variance?

   Take the square root of the variance, $\text{SD}(X) = \text{sqrt}(Var(X))$. Standard deviation is always the positive square root of variance.

10. What is the mean of $X$ if $X$ follows a normal distribution with mean $\text{mu}$?

    $E(X) = \text{mu}$. Normal distribution mean is the location parameter $\mu$.

11. What is the mean of the difference of two independent random variables $X$ and $Y$?

    $E(X-Y) = E(X) - E(Y)$. Expected values subtract when finding difference.

12. Calculate the variance for $X$ with $Var(X) = 4$ after transformation $Y = 3X + 2$.

    $Var(Y) = 36$. Apply variance rule: $Var(Y) = 3^2 \times 4 = 36$.

13. What is the formula for the variance of a linear transformation $Y = a + bX$?

    $Var(Y) = b^2 Var(X)$. Variance multiplied by $b^2$; constant $a$ has no effect.

14. What is the variance of a Poisson random variable with rate $\text{lambda}$?

    $Var(X) = \text{lambda}$. For Poisson distribution, variance equals the rate parameter.

15. Identify the mean of a uniformly distributed random variable $X$ over $a$ to $b$.

    $E(X) = \frac{a+b}{2}$. Uniform mean is the midpoint of the interval.

16. What is the variance of the difference of two independent random variables $X$ and $Y$?

    $Var(X-Y) = Var(X) + Var(Y)$. Variances still add even when subtracting variables.

17. What is the variance of $X$ if $X$ is a geometric random variable with probability $p$?

    $Var(X) = \frac{1-p}{p^2}$. Geometric variance uses $(1-p)$ in numerator, $p^2$ in denominator.

18. What is the effect on the mean if $X$ is shifted by $c$, i.e., $Y = X + c$?

    Mean increases by $c$, $E(Y) = E(X) + c$. Adding constant shifts mean but doesn't change spread.

19. Calculate the variance of $X$ with $Var(X)=16$ after $Y=-X$.

    $Var(Y) = 16$. Multiplying by $-1$ doesn't change variance: $(-1)^2 = 1$.

20. For a random variable $X$, what happens to the standard deviation if $X$ is multiplied by $b$?

    Standard deviation is multiplied by $|b|$. Multiplying by $b$ scales standard deviation by $|b|$.

21. Calculate the mean of $X$ if $X$ is uniform from 1 to 5.

    $E(X) = 3$. Uniform from 1 to 5: mean = $(1+5)/2 = 3$.

22. For $X$ uniform from 2 to 8, calculate the variance.

    Variance is $3$. Uniform variance: $\frac{(8-2)^2}{12} = \frac{36}{12} = 3$.

23. Find the mean of a random variable $X$ given $P(X=1)=0.2$, $P(X=2)=0.5$, $P(X=3)=0.3$.

    $E(X) = 2.1$. Sum each value times its probability: $1(0.2) + 2(0.5) + 3(0.3)$.

24. What is the formula for the expected value of a discrete random variable $X$?

    $E(X) = \text{sum of } (x_i \times p_i)$. Each value times its probability, then sum all products.

25. State the formula for the mean of a linear transformation $Y = a + bX$.

    $E(Y) = a + bE(X)$. Linear transformation: add constant $a$, multiply by $b$.

26. What is the mean of $X$ if $X$ is a geometric random variable with probability $p$?

    $E(X) = \frac{1}{p}$. Geometric mean is reciprocal of success probability.

27. What is the standard deviation of a binomial random variable $X$ with $n$ trials and probability $p$?

    $\text{SD}(X) = \text{sqrt}(n \times p \times (1-p))$. Binomial standard deviation uses $n$, $p$, and $(1-p)$ under square root.

28. Calculate the mean of $X$ with $E(X)=3$ after $Y=-X$.

    $E(Y) = -3$. Multiplying by $-1$ changes sign: $E(-X) = -E(X)$.

29. If $X$ is a random variable with variance 25, what is $Var(2X + 3)$?

    $Var(2X + 3) = 100$. Constant doesn't affect variance: $Var(2X + 3) = 4(25)$.

30. Find the mean of a random variable $X$ with $E(X)=4$ after $Y=2X-3$.

    $E(Y) = 5$. Apply transformation: $E(Y) = 2(4) - 3 = 5$.

31. Identify the standard deviation of a Poisson random variable with rate $\lambda$.

    Standard deviation is $\sqrt{\lambda}$. Poisson standard deviation is square root of rate parameter.

32. What is the mean of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?

    $E(X) = \frac{1}{\text{lambda}}$. Exponential mean is reciprocal of rate parameter.

33. What is the variance of $X$ if $X$ follows an exponential distribution with rate $\text{lambda}$?

    $Var(X) = \frac{1}{\text{lambda}^2}$. Exponential variance is reciprocal of rate squared.

34. If $X$ is a random variable with mean 10, what is $E(2X + 3)$?

    $E(2X + 3) = 23$. Apply linear transformation: $E(2X + 3) = 2(10) + 3$.

35. What is the variance of a uniformly distributed random variable $X$ over $a$ to $b$?

    $Var(X) = \frac{(b-a)^2}{12}$. Uniform variance uses range squared divided by 12.

36. What is the variance of the sum of two independent random variables $X$ and $Y$?

    $Var(X+Y) = Var(X) + Var(Y)$. Variances add for independent variables (never subtract).

37. Identify the standard deviation of $X$ if $X$ follows a normal distribution with variance $\text{sigma}^2$.

    Standard deviation is $\text{sigma}$. Standard deviation is square root of variance: $\sqrt{\sigma^2} = \sigma$.

38. What is the mean of a binomial random variable $X$ with $n$ trials and probability $p$?

    $E(X) = n \times p$. Binomial mean equals number of trials times success probability.

39. What is the mean of the sum of two independent random variables $X$ and $Y$?

    $E(X+Y) = E(X) + E(Y)$. Expected values always add for independent variables.

40. Find the standard deviation for $Var(X)=4$.

    $\text{SD}(X) = 2$. Standard deviation is the square root of variance.

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