Generating...                               quiz05b_n1

  1. The germination time in days of a newly planted seed has a gamma distribution with shape parameter $\alpha = 4$ and rate parameter $\lambda = 2$. If the germination times $X_1,\ldots,X_n$ are independent, estimate the probability that the average germination time $\bar{X}$ of $n = 25$ seeds is between 1.7 and 2.2 days.

    $\Phi( 1.0) - \Phi( -1.5) \approx 0.7745$ $\Phi( 0.58) - \Phi( -1.92) \approx 0.6925$ $\Phi( 0.7) - \Phi( -2.3) \approx 0.7473$ $\Phi( 1.2) - \Phi( -1.8) \approx 0.849$

  2. An actual voltage of new battery has a uniform density function between 5 and 7. Consider the sum $Y$ of the voltages from 90 new batteries. Find $E[Y]$ and $Var(Y)$.

    $E[Y] = 540$ and $Var(Y) = 30$ $E[Y] = 180$ and $Var(Y) = 30$ $E[Y] = 180$ and $Var(Y) = 30$ $E[Y] = 540$ and $Var(Y) = 90$ $E[Y] = 540$ and $Var(Y) = 90$ $E[Y] = 180$ and $Var(Y) = 10$

  3. The germination time in days of a newly planted seed has a gamma distribution with shape parameter $\alpha = 1$ and rate parameter $\lambda = 2$. If the germination times $X_1,\ldots,X_n$ are independent, find $E[\bar{X}]$ and $Var(\bar{X})$ for the average germination time $\bar{X}$ of $n = 100$ seeds.

    $E[\bar{X}] = {{1}\over{2}}$ and $Var(\bar{X}) = {{1}\over{4}}$ $E[\bar{X}] = 2$ and $Var(\bar{X}) = {{1}\over{4}}$ $E[\bar{X}] = 2$ and $Var(\bar{X}) = {{1}\over{400}}$ $E[\bar{X}] = 2$ and $Var(\bar{X}) = {{1}\over{144}}$ $E[\bar{X}] = {{1}\over{2}}$ and $Var(\bar{X}) = {{1}\over{400}}$ $E[\bar{X}] = {{1}\over{2}}$ and $Var(\bar{X}) = {{1}\over{144}}$

  4. Suppose that $X$ has the density function $f(x) = c\,x^2$ for $0\le x\le 2$. Identify the constant $c$, and then find $E[X]$.

    4 $\displaystyle {{8}\over{5}}$ $\displaystyle {{3}\over{2}}$ $\displaystyle {{32}\over{5}}$

  5. Let $X$ be a normal random variable with $\mu = 6$ and $\sigma = 2$. Find the value of $x$ such that $P(X > x) = 0.2$.

    $1.64$ $9.28$ $7.68$ $6.84$ $0.84$ $8.0$

  6. Let $X$ be a normal random variable with $\mu = 6$ and $\sigma = 20$. Find $P( 10 < X < 30)$.

    $0.9772 - 0.5 = 0.4772$ $0.9918 - 0.6554 = 0.3364$ $0.8413 - 0.5 = 0.3413$

    $0.8849 - 0.5793 = 0.3057$

  7. Suppose that $X$ has the density function $f(x) = c\,x^3$ for $0\le x\le 1$. Identify the constant $c$, and then express $P(a \le X \le b)$ in terms of $a$ and $b$ if $0\le a< b\le 1$.

    $P(a \le X \le b) = 2\,b - 2\,a$ $P(a \le X \le b) = 4\,b^3 - 4\,a^3$ $P(a \le X \le b) = b^2 - a^2$ $P(a \le X \le b) = b^4 - a^4$

  8. An actual voltage $X$ of new battery has a uniform density function between 1 and 5. Find $E[X]$ and $Var(X)$.

    $E[X] = 1$ and $Var(X) = {{4}\over{3}}$ $E[X] = 3$ and $Var(X) = 4$ $E[X] = 1$ and $Var(X) = 2$ $E[X] = 3$ and $Var(X) = {{4}\over{3}}$ $E[X] = 3$ and $Var(X) = 2$ $E[X] = 1$ and $Var(X) = 4$

  9. Suppose that the lifetime of an electronic component follows an exponential distribution with rate parameter $\lambda = {{1}\over{5}}$. Find the probability that the lifetime is between 15 and 25.

    $\displaystyle e^ {- 10 }$ $\displaystyle e^ {- 15 }-e^ {- 25 }$ $\displaystyle e^ {- 2 }$ $\displaystyle e^ {- 3 }-e^ {- 5 }$

  10. Suppose that $X$ has the density function $f(x) = c\,x^3$ for $0\le x\le 1$. Identify the constant $c$, and then find the cdf $F(x)$.

    $F(x) = x^4$ for $0\le x\le 1$ $F(x) = 4\,x^3$ for $0\le x\le 1$ $F(x) = x^3$ for $0\le x\le 1$ $F(x) = 3\,x^2$ for $0\le x\le 1$


Department of Mathematics
Last modified: 2025-05-04