Generating...                               quiz05b_n15

  1. Suppose that $X$ has the density function $f(x) = c\,x^3$ for $0\le x\le 2$. Identify the constant $c$, and then find $E[X]$.

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

  2. Let $X$ be a normal random variable with $\mu = 10$ and $\sigma = 2$. Find the value of $x$ such that $P(X > x) = 0.05$.

    $1.64$ $26.4$ $12.56$ $13.28$ $1.28$ $30.0$

  3. The germination time in days of a newly planted seed has a gamma distribution with shape parameter $\alpha = 9$ and rate parameter $\lambda = {{1}\over{2}}$. 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 $n = 36$ seeds.

    $E[\bar{X}] = {{9}\over{2}}$ and $Var(\bar{X}) = 36$ $E[\bar{X}] = {{9}\over{2}}$ and $Var(\bar{X}) = {{9}\over{25}}$ $E[\bar{X}] = 18$ and $Var(\bar{X}) = 36$ $E[\bar{X}] = 18$ and $Var(\bar{X}) = {{9}\over{25}}$ $E[\bar{X}] = 18$ and $Var(\bar{X}) = 1$ $E[\bar{X}] = {{9}\over{2}}$ and $Var(\bar{X}) = 1$

  4. Let $X$ be a normal random variable with $\mu = 2$ and $\sigma = 10$. Find $P( 10 < X < 25)$.

    $0.9893 - 0.7881 = 0.2011$ $0.9713 - 0.6554 = 0.3159$

    $0.8749 - 0.6554 = 0.2195$ $0.8289 - 0.5793 = 0.2497$

  5. An actual voltage of new battery has a uniform density function between 1 and 5. Consider the sum $Y$ of the voltages from 90 new batteries. Find $E[Y]$ and $Var(Y)$.

    $E[Y] = 90$ and $Var(Y) = 30$ $E[Y] = 270$ and $Var(Y) = 120$ $E[Y] = 270$ and $Var(Y) = 90$ $E[Y] = 90$ and $Var(Y) = 40$ $E[Y] = 90$ and $Var(Y) = 60$ $E[Y] = 270$ and $Var(Y) = 180$

  6. The germination time in days of a newly planted seed has a gamma distribution with shape parameter $\alpha = 9$ 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 $n = 100$ seeds is between 4.8 and 5.1 days.

    $\Phi( 1.5) - \Phi( 0.5) \approx 0.2417$ $\Phi( 4.0) - \Phi( 2.0) \approx 0.0227$ $\Phi( 2.0) - \Phi( 1.0) \approx 0.1359$ $\Phi( 3.0) - \Phi( 1.0) \approx 0.1573$

  7. An actual voltage of new battery has a uniform density function between 5 and 7. Consider the sum $Y$ of the voltages from 48 new batteries. Approximate $P( 285 < Y < 291)$.

    $\Phi( {{3}\over{4}}) - \Phi( -{{3}\over{4}}) \approx 0.5467$ $\Phi( 3) - \Phi( -3) \approx 0.9973$ $\Phi( 1) - \Phi( -1) \approx 0.6827$ $\Phi( {{4}\over{3}}) - \Phi( -{{4}\over{3}}) \approx 0.8176$

  8. An actual voltage $X$ of new battery has a uniform density function between 4 and 8. Find $E[X]$ and $Var(X)$.

    $E[X] = 6$ and $Var(X) = {{4}\over{3}}$ $E[X] = 4$ and $Var(X) = 2$ $E[X] = 4$ and $Var(X) = {{4}\over{3}}$ $E[X] = 6$ and $Var(X) = 2$ $E[X] = 6$ and $Var(X) = 4$ $E[X] = 4$ 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 10 and 15.

    $\displaystyle e^ {- 2 }-e^ {- 3 }$ $\displaystyle e^ {- 10 }-e^ {- 15 }$ $\displaystyle e^ {- 1 }$ $\displaystyle e^ {- 5 }$

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

    $F(x) = {{2\,x}\over{9}}$ for $0\le x\le 3$ $F(x) = {{x^3}\over{27}}$ for $0\le x\le 3$ $F(x) = {{x^2}\over{9}}$ for $0\le x\le 3$ $F(x) = {{x^2}\over{9}}$ for $0\le x\le 3$


Department of Mathematics
Last modified: 2025-06-19